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UNIVERSITY     OF     ILLINOIS     BULLETIN 

Issued  Weekly 
Vol.  XXIII  October  12,  1925  No.  6 

[Entered  as  second-class  matter  December  II,  1912,  at  the  post  office  at  Urbana,  Illinois,  under  the 
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EDUCATIONAL   RESEARCH  CIRCULAR  NO.   37 


BUREAU  OF  EDUCATIONAL  RESEARCH 
COLLEGE  OF  EDUCATION 

HOW  TO  MAKE  A  COURSE  OF  STUDY 
IN  ARITHMETIC 

By 

M.  E.  Herriott 

Associate,   Bureau  of  Educational  Research 


IE  II  OF  THf 


JAN     o  19?6 


PUBLISHED  BY  THE  UNIVERSITY  OF  ILLINOIS 
URBANA 


Digitized  by  the  Internet  Archive 

in  2012  with  funding  from 

University  of  Illinois  Urbana-Champaign 


http://www.archive.org/details/howtomakecourseo37herr 


-LJL  6e~ 

HOW  TO  MAKE  A  COURSE  OF  STUDY 

IN  ARITHMETIC 

INTRODUCTION 

Purpose  of  circular.  The  purpose  of  this  circular  is  to  indicate 
the  application  to  arithmetic  of  the  general  technique  of  course-of- 
study  making.1  The  circular  is  intended  to  furnish  a  working  basis 
and  a  guide  to  those  who  are  making  or  revising  the  course  of  study 
in  arithmetic.2 

Plan  of  circular.  In  order  to  provide  a  general  view  of  the  con- 
tent and  organization  of  a  course  of  study  in  arithmetic,  the  outline 
for  an  arithmetic  course  of  study  is  given  first.  The  different  divi- 
sions of  this  outline  are  then  considered  in  detail;  the  nature  of  the 
content  for  each  division  is  discussed,  and  suggestions  are  made 
regarding  the  procedure  to  be  followed.  The  appendix  includes  three 
types  of  material:  first,  the  fourth-grade  portion  of  a  course  of  study 
in  arithmetic,  which  was  prepared  to  illustrate  the  application  of  the 
principles  outlined  in  this  circular;  second,  standards  of  attainment 
as  derived  from  standardized  tests  in  the  fundamental  operations  of 
arithmetic;  and  third,  selected  and  annotated  bibliography,  which 
contains  references  valuable  to  those  formulating  courses  of  study  in 
arithmetic. 

Course-of-study  making  a  cooperative  enterprise.  The  making 
of  courses  of  study  is  generally  undertaken  as  a  cooperative  enter- 
prise. When  several  or  all  of  the  courses  of  study  for  a  school  system 
are  being  formulated,  the  teachers  are  usually  organized  into  subject 
committees.  In  addition  there  are  one  or  two  general  committees  to 
oversee  the  whole  work  of  course-of-study  making.  They  give  direc- 
tions and  inspiration  to  the  subject  committees  and  also  perform 
some  of  the  general  work,  such  as  specifying  the  grade  time  allot- 
ments for  the  various  subjects.   The  subject  committees  prepare  the 


aThe  general  principles  for  making  a  course  of  study  are  given  in  a  previous 
circular: 

Monroe,  Walter  S.  "Making  a  course  of  study."  University  of  Illinois 
Bulletin  Vol.  23,  No.  2.  Bureau  of  Educational  Research  Circular  No.  35.  Urbana: 
University  of  Illinois,  1925.     36  p. 

2See  Ibid.,  p.  3-4,  for  the  distinction  between  curriculum  and  course  of  study,  a 
distinction  which  should  be  kept  clear  at  all  times. 

[  3  ] 


courses  of  study  for  the  several  school  subjects.  This  circular 
describes  in  detail  the  task  of  the  subject  committee  on  arithmetic. 
Many  superintendents  state  that  this  cooperative  plan  of  course-of- 
study  making  has  proved  the  most  valuable  work  they  have  under- 
taken for  the  improvement  of  teachers  in  service.3 


3For  a  more  detailed  discussion  of  organizing  for  course  of  study  making  and 
the  benefits  of  such  concerted  effort,  see:  Monroe,  op.  cit. 

[  4  ] 


THE  GENERAL  OUTLINE  OF  AN  ARITHMETIC 
COURSE  OF  STUDY 

The  course  of  study  in  arithmetic  should  contain  two  types  of 
material:  first,  specifications,  which  include  objectives  (general  and 
specific),  organization  of  topics,  and  time  allotments;  second, 
directions  relative  to  instruction,  which  include  suggestions  as  to 
learning  exercises  and  methods  of  teaching. 

Outline  for  a  course  of  study  in  arithmetic.  The  following  out- 
line is  suggested  as  a  working  basis  for  an  arithmetic  course  of  study 
covering  the  work  of  six  grades: 
I.  Introduction 

1.  Purpose  of  this  course  of  study 

2.  The  general  objectives  of  arithmetic 

3.  General  organization  of  the  course  in  arithmetic 

A.  Grade  time  allotments 

B.  The  sequence  of  specific  objectives  from  grade  to  grade 

(Presented  in  tabular  form) 

C.  The  sequence  of  topics  from  grade  to  grade 

(Presented  in  tabular  form)4 
II.  Course  of  Study  by  Grades 

1.  First  grade 

A.  Specific  objectives 

B.  Suggestions  relative  to  instruction 

a.  Learning  exercises 

b.  Methods 

2.  Second  grade  (subdivisions  as  for  first  grade) 

3.  Third  grade  (subdivisions  as  for  first  grade) 

4.  Fourth  grade 
A.  Specifications 

a.  Specific  objectives 

b.  Topics 

(1)  Organization  of  topics 

(2)  Suggested  time  allotments  for  topics 


This  has  been  effectively  done  in  the  elementary  course  of  study  for  Baltimore. 
"Arithmetic — course  of  study  for  grades  four,  five,  and  six."  City  of  Baltimore, 
Maryland:  Department  of  Education,  1924.  ''Tabulation  of  the  course  of  study 
in  arithmetic.  Kindergarten,  primary  and  intermediate  grades"  (inserted  inside  the 
back  cover). 

[  5  ] 


B.  Directions  relative  to  instruction 

a.  Learning  exercises 

b.  Methods 

5.  Fifth  grade  (subdivisions  as  for  fourth  grade) 

6.  Sixth  grade5  (subdivisions  as  for  fourth  grade) 

III.  References  for  the  Teacher 

It  should  be  noted  that  in  this  outline  there  are  essentially  three 
organizations.  First,  there  is  the  general  organization  of  the  course 
of  study  as  a  whole,  containing  three  divisions:  introduction,  course 
of  study  by  grades,  and  references  for  the  teacher  (indicated  by 
Roman  numerals  I,  II,  and  III).  The  second  organization  is  the 
logical  arrangement  of  the  course  in  arithmetic,  including  grade  time 
allotments,  specific  objectives  and  topics  (placed  in  Introduction). 
Finally,  the  content  of  the  course  of  study  within  each  grade  is 
organized  into  a  form  that  is  usable  by  the  teacher,  giving  first  the 
grade  specifications  and  then  the  directions  for  carrying  them  out. 


5Because  of  the  general  development  of  the  junior-high-school  movement,  this 
outline  ends  with  the  sixth  grade.  If  arithmetic  is  continued  through  the  seventh 
and  eighth  grades,  the  outline  may  be  continued  in  its  present  form.  If  a  "unified 
mathematics"  course  is  offered  above  the  sixth  grade,  a  somewhat  different  treat- 
ment may  be  needed. 

[  6  ] 


I.  THE  INTRODUCTION  TO  AN  ARITHMETIC 
COURSE  OF  STUDY 

Necessity  for  a  point  of  view.  The  introductory  section  of  a 
course  of  study  should  present  in  definite  terms  the  point  of  view 
which  is  exemplified  in  the  later  sections.  In  other  words,  the  intro- 
duction should  present  briefly  the  author's  educational  philosophy  as 
applied  to  arithmetic.  This  will  include  some  such  topics  as  the 
following:  the  purpose  of  the  course  of  study,  the  nature  of  the 
learning  and  teaching  processes  in  arithmetic,  the  use  of  the  text- 
book in  arithmetic,  and  the  purposes  of  teaching  arithmetic.  The 
last  topic  leads  directly  into  the  discussion  of  general  objectives. 

General  objectives.  The  principal  function  of  the  discussion  of 
general  objectives  of  arithmetic  is  to  furnish  the  background  for  the 
consideration  of  the  detailed  objectives.  The  following  statements 
are  typical  of  those  used  in  expressing  general  objectives  in 
arithmetic.6 

1.  To  give  arithmetical  knowledge  that  fits  into  "real  situa- 
tions" in  the  school,  home,  shop,  or  social  life  of  the  pupils  and 
to  prepare  them  to  meet  similar  situations  in  adult  life. 

2.  To  give  power  to  use  mental  calculations  in  every-day 
demands. 

3.  To  train  in  proper  habits  of  accuracy,  speed,  neatness, 
and  checking  results  in  mathematical  problems. 

4.  To  give  information  about  the  conduct  of  every-day 
business. 

5.  To  develop  habits  of  logical  reasoning  from  conditions 
to  results. 

Formulating  the  point  of  view  and  general  objectives.  No 
doubt  anyone  who  starts  to  write  a  course  of  study  in  arithmetic  will 
have  already  an  educational  philosophy  and  a  set  of  general 
objectives,  but  neither  may  be  clearly  formulated.  In  such  instances, 
it  would  be  well  to  read  one  or  two  books  on  educational  theory  and 
methods  of  teaching  arithmetic  before  attempting  to  write  out  a 
point  of  view  and  general  objectives.7    With  a  general  educational 


""Arithmetic — elementary  course  of  study."    Trenton:   New  Jersey:    Board  of 
Education,  1923,  p.  9 

7See  bibliography  for  suggested  references. 

[7] 


philosophy  toward  arithmetic  fairly  well  established,  the  writers  of  a 
course  of  study  are  ready  to  proceed  with  the  other  tasks  indicated 
in  the  outline  on  pages  5-6.  They  may  find  later  that  their  point  of 
view  has  changed  in  certain  details  and  that  they  wish  to  rewrite 
portions  of  what  they  have  formulated,  but  they  need  these  first 
statements  to  guide  them  as  they  develop  the  course  of  study. 

Organization  of  the  course  in  arithmetic.  'The  organization  of 
the  course  in  arithmetic"  should  be  prepared  after  the  point  of  view 
and  general  objectives  have  been  formulated  and  before  the  directions 
relative  to  instruction  in  the  separate  grades  are  written.  This  organ- 
ization consists  of  three  things:  first,  the  grade  time  allotments; 
second,  the  sequence  of  the  specific  objectives  from  grade  to  grade; 
and  third,  the  sequence  of  topics  from  grade  to  grade. 

Grade  time  allotments.  The  grade  time  allotments  for  arithmetic 
probably  will  have  been  determined  in  advance  by  a  general  com- 
mittee8 and  will  need  only  to  be  accepted  and  adjusted  harmoniously 
in  the  course  of  study.  Time  allotments  have  not  been  scientifically 
determined,  but  the  usual  practices  have  been  made  the  subject  of 
study.  In  an  investigation9  made  in  forty-nine  cities,  the  following 
grade  time  allotments  to  arithmetic  in  terms  of  minutes  per  week  are 
given: 


Grades 

I 

n 

in 

IV                V 

VI 

VII 

VIII 

96 

143 

193 

206       211 

211 

212 

211 

Minutes  per  week 

No.  of  cities  giving  arith- 
metic       32         48         48         48         48         48  42       36 

This  table  should  be  read  as  follows:  thirty-two  cities,  out  of 
forty-nine  studied,  teach  arithmetic  in  Grade  I  on  an  average  of 
ninety-six  minutes  per  week,  and  so  forth.  The  average  time  devoted 
to  arithmetic  in  the  forty-nine  cities  is  1,451  minutes  per  week. 

Sequence  of  specific  objectives  and  topics  from  grade  to  grade.10 
The  course  in  arithmetic  should  not  be  considered  in  a  piecemeal 
fashion,  for  instance,  for  one  grade  in  isolation  from  other  grades; 
but  it  should  be  taken  as  a  whole  for  the  elementary  school.   In  order 


sSee  page  3  for  cooperative  organization  for  course-of-study  making. 

9"Facts  on  the  public  school  curriculum."  Research  Bulletin  of  the  National 
Education  Association,  Vol.  I,  No.  5.  Washington:  Research  Division  of  the 
National  Education  Association,  1923,  p.  326-27. 

^'Specific  objectives  and  topics  are  discussed  at  length  later,  p.  10-14.  The 
purpose  here  is  only  to  point  out  the  manner  of  presenting  them  in  the  introductory 
section  of  the  course  of  study. 

[  8  ] 


to  provide  this  unified  view  of  the  course,  the  sequence  of  specific 
objectives  and  of  topics  from  grade  to  grade  should  be  presented  in 
the  introductory  section  of  the  course  of  study.  Probably  such  a  pre- 
sentation can  be  made  most  effective  when  given  in  tabular  form.11 
An  illustration  of  the  beginning  of  such  a  form  is  given  below: 

SEQUENCE    OF    TOPICS    IN    ELEMENTARY    ARITHMETIC  . 


Grade 

Counting 

Reading  and  Writing  Number 

Etc. 

Arabic 

Roman 

I 

II 

III 

Etc. 

This  same  form  should  be  used  for  the  specific  objectives  so  that 
they  also  may  be  presented  as  a  unified  whole. 

Both  of  these  organizations  form  a  summary  of  the  specifications 
as  formulated  for  the  various  grades.  Although  they  are  given  in  the 
introductory  section  of  the  course  of  study,  they  cannot  be  prepared 
until  the  specifications  for  each  grade  have  been  formulated. 


UA  good  example  of  this  may  be  found  in  the  elementary  course  of  study 
for  Denver,  Colorado: 

"Arithmetic — grades  1,  2,  3,  4,  5,  and  6 — elementary  school,"  Denver:  Board 
of  Education,  1924,   (insert  inside  front  cover). 

[9] 


II.  COURSE  OF  STUDY  BY  GRADES 

General  content  of  course  of  study  by  grades.  As  was  noted, 
page  5,  the  arithmetic  course  of  study  by  grades  should  contain  two 
types  of  material:  first,  specifications;  second,  directions  relative  to 
instruction.  The  specifications  include  time  allotments,  specific 
objectives,  and  topics,  and  should  be  presented  at  the  beginning  of 
the  portion  of  the  course  of  study  for  each  grade.  The  suggestions  on 
how  to  direct  the  pupils  in  their  learning  so  as  to  accomplish  the 
objectives  should  be  very  specific  and  should  pertain  directly  to  the 
specifications  for  each  grade.  The  outlines  for  the  first  three  grades, 
ac  given  on  page  5,  are  less  elaborate  than  those  for  the  later  ones, 
because  of  the  informality  of  the  work  in  the  lower  grades.12 

General  nature  of  specifications.  The  grade  specifications  in- 
clude specific  objectives,  organization  of  topics,  and  time  allotments. 
The  last  needs  no  discussion  here  because  it  is  treated  in  detail 
on  page  8. 

Nature  of  specific  objectives.  Specific  objectives,  as  the  name 
implies,  are  particularized  statements  of  general  objectives.  They 
should  be  as  detailed  and  definite  as  possible  and  should  be  expressed 
in  terms  of  ability  to  do.  Whenever  feasible,  they  should  specify  the 
degrees  of  the  abilities  to  be  attained.  For  example,  a  specific 
objective  in  both  fifth  and  sixth-grade  arithmetic  might  be  "the 
ability  to  multiply  four-place  numbers  by  two-place  numbers,"  but 
this  statement  is  still  of  too  general  a  nature  to  differentiate  between 
the  fifth  and  sixth-grade  standards.  The  degree  of  the  ability  to  be 
attained  should  be  added.  Thus,  in  the  fifth  grade  the  pupils  should 
attain  "the  ability  to  multiply  four-place  numbers  by  two-place 
numbers  at  the  rate  of  eight  examples  in  eight  minutes,"  while  in  the 
sixth  grade  the  rate  should  be  "nine  examples  in  six  minutes." 

Whenever  standards  of  this  nature  have  been  worked  out,  they 
should  be  used  by  the  course-of-study  writers.  However,  there  are 
many  other  abilities  to  be  engendered  by  arithmetic  for  which  degrees 
of  attainment  cannot  be  so  definitely  stated,  as  for  example,  in  the 
following  objectives:    In  Grade  I,  "the  ability  to  read  numbers  to 


^See  footnote,  page  12,  for  suggestions  relative  to  the  time  for  beginning  formal 
work  in  arithmetic. 

[10] 


100;"  in  Grade  II,  "the  ability  to  write  numbers  to  1,000;"  in 
Grade  VI,  "the  ability  to  originate  problems,"  or  "the  ability  to  draw 
to  scale."  Even  though  the  degrees  of  attaniment  in  such  abilities 
have  not  been  worked  out,  and  may  never  be  in  all  cases,  they 
should  be  stated  in  as  definite  terms  as  possible.13  General  terms 
such  as  "proficiency,"  "fluently,"  and  "quickly"  and  statements  in 
terms  of  ground  to  be  covered  or  of  so  many  pages  in  the  textbook 
should  be  avoided. 

How  to  formulate  specific  objectives.  There  are  several  sources 
from  which  specific  objectives  may  be  obtained:  first,  arithmetic 
textbooks;14  second,  books  on  methods  of  teaching  arithmetic;  third, 
numerous  courses  of  study;  fourth,  various  kinds  of  standardized 
tests  in  arithmetic;  and  finally,  special  studies  such  as  the  one  made 
in  Iowa  by  Wilson.15  The  information  of  those  who  formulate  the 
objectives  may  be  considered  as  another  source,  but  their  knowledge 
and  judgment  should  be  used  more  for  the  purpose  of  modifying 
and  adjusting  the  objectives  found  elsewhere.  This  function  of  judg- 
ment needs  to  be  performed  generously. 

After  the  specific  objectives  have  been  collected  from  as  wide  a 
range  of  sources  as  possible,  they  must  be  culled  or  selected.  A 
certain  amount  of  rejecting  may  go  on  when  they  are  being  listed, 
but  in  order  that  they  may  be  seen  as  a  whole  and  in  their  proper 
relations  to  each  other,  it  is  probably  better  to  avoid  making  the 
selections  until  a  rather  complete  list  has  been  made  up.  At  this 
point,  for  the  first  time,  the  judgment  of  the  curriculum  builder  enters 
to  an  appreciable  extent.  He  should  be  guided  by  principles  pre- 
viously decided  upon  which  are  flexible  and  allow  for  a  liberal  inter- 
pretation and  yet  definite  enough  to  give  assistance.  Examples  of 
principles  that  are  really  helpful  are  given  in  the  following 
statements:16 


13For  illustrations  of  ^specific  objectives  see  the  appendix:    p.  23-26. 

"In  general,  textbooks  only  imply  objectives  to  be  attained,  although  some 
of  the  newer  arithmetics  are  including  standards  of  attainment.    For  example: 

Watson,  Bruce  M.  and  White,  Charles  E.  Modern  Intermediate  Arith- 
metic.     Boston:  D.  C.  Heath  and  Company,  1922.     254  p. 

15W'ilson,  Guy  M.  "A  survey  of  the  social  and  business  uses  of  arithmetic." 
Sixteenth  Yearbook  of  the  National  Society  for  the  Study  of  Education,  Part  I. 
Bloomington,  Illinois:    Public  School  Publishing  Company,  1917,  p.  128-42. 

16Caldwell,  Otis  W.  "Types  and  principles  of  curricular  development," 
Teachers  College  Record,  24:326-37,  September,  1923. 

[11] 


1.  Mathematics,  in  the  elementary  and  junior-high-school 
grades,  should  be  primarily  a  tool  for  the  quantitative  thinking 
that  children  and  adults  need  to  do. 

2.  Each  year  should  give  the  most  intrinsically  valuable 
mathematical  information  and  training  which  the  pupil  is 
capable  of  receiving  at  that  time,  with  little  consideration  of  the 
needs  of  subsequent  courses. 

3.  This  aim  necessitates  the  inclusion  in  junior-high-school 
grades  of  certain  elements  of  arithmetic,  intuitive  geometry, 
algebra,  trigonometry,  and  statistics,  although  these  are  not  to 
be  rigidly  classified  under  the  traditional  divisions  as  named. 

4.  Manipulation  of  mathematical  symbols  as  an  end  should 
be  omitted. 

5.  Attention  should  be  directed  toward  a  better  appreciation 
of  the  part  that  mathematics  has  occupied  and  is  now  occupy- 
ing in  the  progress  of  civilization. 

6.  There  should  be  a  marked  increase  *in  the  accuracy  of 
computation  with  integers,  fractions,  and  percents. 
Organization  of  specific  objectives.    The  specific  objectives  of 

arithmetic  need  to  be  organized  in  proper  sequence,  so  that  their 
realization  will  mean  a  gradual  yet  sure  development  of  arithmetical 
ability  on  the  part  of  the  pupils.  While  the  objectives  are  being  col- 
lected, they  will  no  doubt  be  assigned  tentatively  to  particular 
grades;  but  after  a  fairly  complete  list  has  been  made,  they  will 
need  to  be  gone  over  carefully  and  reorganized  so  as  to  assure  a  uni- 
fied whole.17  A  few  guiding  principles,  such  as  the  following,  should 
be  formulated  before  making  the  reorganization: 

1.  Provisions  must  be  made  usually  for  the  development  of 
an  ability  over  a  period  of  more  than  one  year. 

2.  The  introduction  of  totally  new  objectives  should  not  be 
too  rapid. 

3.  The  organization  should  involve  a  "psychological"  rather 
than  a  strictly  "logical"  sequence. 


170ne  of  the  largest  factors  in  determining  the  grade  placement  of  objectives 
is  the  grade  in  which  formal  work  in  arithmetic  is  begun.  It  seems. that  no  formal 
work  should  be  given  in  the  first  grade,  probably  none  in  the  second  and  very  little 
in  the  third.    For  a  detailed  discussion  see: 

Wilson,  Guy  M.  "Arithmetic,"  Third  Yearbook  of  the  Department  of 
Superintendence.  Washington:  Department  of  Superintendence  of  the  National 
Education  Association,  1925,  p.  37-40. 

[12] 


Presentation  of  specific  objectives.  The  specific  objectives 
should  be  presented  not  only  at  the  beginning  of  the  course  of  study 
for  each  grade,  but  also  in  the  introductory  section  of  the  course  of 
study,  as  suggested  on  page  9. 

Relation  of  specific  objectives  to  topics.  It  is  best  to  complete 
the  organization  of  the  specific  objectives  before  outlining  the  topics, 
for  theoretically  the  specific  objectives  should  guide  the  latter  task. 
However,  in  planning  an  outline  of  topics  for  a  year,  some  new 
objectives  probably  will  be  suggested.  In  order  to  insure  that  the 
outline  is  compatible  with  the  list  of  objectives,  a  careful  comparison 
should  be  made.  The  writer  of  a  course  of  study  should  make  certain 
also  that  all  topics  implied  in  the  list  of  objectives  have  been  included 
in  the  outline. 

Selection  of  topics.  The  sources  of  suggestions  for  the  topics  of 
arithmetic  are  much  the  same  as  for  specific  objectives:  textbooks, 
books  on  methods  of  teaching  arithmetic,  courses  of  study,  standard- 
ized tests,  and  special  studies.18  Chief  among  these  are  textbooks, 
courses  of  study,  and  special  studies.  These  three  sources  should  be 
thoroughly  canvassed  and  the  topics  outlined  as  much  in  detail  as 
possible.  It  is  also  well  to  make  approximate  grade  placements 
before  making  comparisons  with  the  specific  objectives. 

Organization  of  topics.  In  the  organization  of  topics,  the  spe- 
cific objectives  and  the  topics  should  be  compared  and  additions  or 
subtractions  made  in  accordance  with  the  results  of  scientific  studies, 
with  the  judgments  of  those  who  are  writing  the  course  of  study, 
and  with  principles  previously  set  up.  At  the  same  time,  the  final 
organization  of  the  topics  and  objectives  in  relation  to  each  other 
should  be  made.  As  any  organization  that  is  determined  upon  must 
be  coordinated  with  the  textbooks  to  be  used,  due  consideration 
should  be  given  to  the  arrangement  and  development  of  the  topics  in 
the  text. 

When  this  organization  has  been  completed,  the  topics  should  be 
arranged  in  tabular  form  as  suggested  on  page  9,  in  order  that  they 
may  be  seen  as  a  whole  by  each  teacher.  Such  a  table  presents 
graphically  the  relation  of  the  work  in  each  grade  to  that  of  all  the 


18For  a  digest  of  the  special  studies  that  have  been  made  on  the  arithmetic 
curriculum  see: 

Wilson,  Guy  M.  "Arithmetic,"  Third  Yearbook  of  the  Department  of  Super- 
intendence. Washington:  Department  of  Superintendence  of  the  National  Education 
Association,  1925,  p.  35-109. 

[13] 


other  grades,  and  the  relation  of  the  specific  objectives  to  the  topics. 
With  these  two  points  of  view,  a  teacher's  work  should  be  much 
more  effective  than  if  she  saw  only  the  work  of  her  grade  in  isolation. 

Provisions  for  individual  differences  by  modifications  of  spe- 
cific objectives  and  topics.19  In  formulating  specific  objectives  and 
in  selecting  topics,  consideration  should  be  given  to  provisions  for 
individual  differences.  Some  eliminations  may  be  made  for  the  slower 
pupils  and  some  additions  for  the  brighter  ones.  All  such  provisions 
should  be  carefully  considered  and  not  decided  upon  hastily.  If  the 
school  is  organized  into  "X,  Y,  Z"  ability  groups,  greater  variations 
may  be  made  than  if  there  is  no  segregation.  However,  most  of  the 
provisions  for  individual  differences,  especially  in  the  latter  instance 
when  homogeneous  grouping  is  not  attempted,  must  be  made  in 
adjusting  learning  exercises  and  methods  of  instruction.20 

General  nature  of  suggestions  relative  to  instruction.  Sugges- 
tions relative  to  instruction  fall  into  two  classes;  those  relating  to 
learning  exercises,  and  those  to  methods  of  instruction. 

Suggestions  for  learning  exercises.  Arithmetic  textbooks  are 
chiefly  compilations  of  learning  exercises,  but  teachers  of  arithmetic 
must  devise  many  additional  exercises  in  order  that  pupils  may 
achieve  the  objectives  set  for  them.  Especially  is  this  true  in  teach- 
ing and  solving  problems  that  involve  new  words.  Additional  reading 
exercises  are  often  needed,  explanations  by  the  teacher  are  necessary, 
or  the  pupils  may  be  required  to  make  diagrams  or  to  do  many  other 
types  of  exercises  not  given  in  the  textbook. 

The  process  of  long  division  may  furnish  a  detailed  example  of 
the  need  for  formulating  and  assigning  appropriate  learning  exercises 
and  of  the  assistance  that  may  be  given  the  teacher  by  the  course  of 
study.  In  achieving  the  objective,  "ability  to  do  long-division 
examples  of  the  type  25)  6775  at  the  rate  of  four  in  eight  minutes," 
which  is  an  appropriate  objective  for  a  fourth-grade  class  to  achieve 
at  the  end  of  the  year,  there  are  a  variety  of  learning  exercises  that 
the  teacher  might  set  up  for  the  pupils  to  do.  Suitable  learning 
exercises  when  the  process  is  first  being  learned  are  different  from 
appropriate  ones  when  long  division  is  understood  as  a  process  but 


"Two  courses  of  study  which  make  very  definite  subject-matter  provisions  for 
individual  differences  are  the  elementary  course  of  study  in  arithmetic  for  Long 
Beach,  California,  and  that  for  Trenton,  New  Jersey. 

^Modifications  of  learning  exercises  and  methods  of  instruction  are  discussed 
later,  pages  IS,  20;  34-36. 

[14] 


when  the  ability  to  perform  the  operations  with  sufficient  rapidity 
has  not  been  attained.  In  formulating  the  learning  exercises,  care 
must  be  taken  to  avoid  having  the  pupil  meet  too  many  difficulties 
at  first.  Osburn21  points  out  that  there  are  four  centers  of  trouble  in 
learning  to  do  long  division:  "(1)  in  getting  acquainted  with  the  new 
form,  (2)  in  carrying,  (3)  in  borrowing,  and  (4)  in  estimating  the 
quotient  including  the  use  of  zeros."  It  is  important  that  the  learning 
exercises  provide  for  the  mastery  of  all  these  difficulties  yet  not 
introduce  them  all  at  the  beginning.  Osburn  cites  one  book  that 
introduces  the  child  to  long  division  through  the  problem  15)  240, 
in  which  all  the  difficulties  mentioned  above  are  encountered. 

After  the  pupils  understand  long  division  as  a  process  but  do 
not  have  the  ability  to  do  such  exercises  with  sufficient  rapidity,  drill 
exercises  that  are  of  a  different  nature  from  the  earlier  learning 
exercises  should  be  given.  In  order  to  engender  a  complete  mastery 
of  the  process,  teachers  should  assign  as  learning  exercises  problems 
that  involve  long  division  but  in  which  the  pupil  must  decide  when 
and  when  not  to  use  the  process.  Such  problems  would  be  entirely 
inappropriate  until  the  pupils  had  a  thorough  understanding  of  long 
division  as  a  process. 

In  the  part  of  the  course  of  study  that  deals  with  the  fourth 
grade,  the  teacher's  attention  should  be  called  to  the  appropriateness 
and  inappropriateness  of  different  types  of  learning  exercises  which 
might  be  used  in  teaching  long  division,  and  suggestions  should  be 
made  as  to  those  types  of  exercises  which  may  be  used  with  the 
greatest  effectiveness.  If  the  textbook  happens  to  be  weak  in  its 
method  of  handling  long  division,  the  course  of  study  can  be 
especially  helpful  to  the  teacher. 

It  should  be  borne  in  mind  by  those  who  write  the  course  of 
study  that  learning  is  an  active  process,  that  the  pupil  learns  only 
through  his  own  activity,  physical  or  mental,  and  that  there  is  no 
such  thing  as  a  pouring-in  process  or  an  inscribing  on  a  blank  tablet. 
Effective  activity  on  the  part  of  the  pupil  must  be  provided  for  by 
the  teacher. 

Effect  of  interests  and  local  conditions  on  learning  exercises. 

In  educational  writings  and  discussions,  much  attention  is  given  to  a 
consideration  of  local  conditions  and  interests  in  making  curricula 
and  in  writing  courses  of  study.   There  is  opportunity  in  formulating 


^Osburn,  Worth  J.    Corrective  Arithmetic.    Boston:    Houghton  Mifflin  Com- 
pany, 1924,  p.  68-69. 

[15] 


learning  exercises  not  only  to  provide  for  peculiarities  of  needs  due 
to  local  conditions  and  interests  but  to  take  advantage  of  them.  A 
water  reservoir,  a  dam,  or  a  factory  may  furnish  admirable  material 
for  learning  exercises  that  would  not  be  effective  where  these  did  not 
exist.  All  communities  have  conditions  that  are  peculiar  to  them  and 
of  which  advantage  can  be  taken.  It  is  possible  that  the  same 
objectives  and  curriculum  will  function  as  well  in  any  one  of  a  large 
number  of  communities  as  in  any  other,  but  it  is  easily  conceivable 
that  many  significant  local  differences  exist  which  may  be  pertinent 
and  valuable  in  formulating  learning  exercises.22 

General  nature  of  suggestions  on  methods  of  instruction.23  The 
course  of  study  performs  a  supervisory  function.  Its  purpose  is  to 
help  and  to  coordinate  the  work  of  the  teachers  of  a  school  system. 
Some  teachers  have  a  well-developed  technique  as  a  result  of  train- 
ing and  experience;  others  are  relatively  inexperienced  and  have 
little  training;  others  have  not  taught  in  the  particular  grade  in 
which  they  are  now  teaching;  and  still  others  are  new  to  the  school 
system,  although  experienced  and  well-trained.  The  course  of  study 
should  help  all  these  teachers  to  use  appropriate  methods  in 
particular  grades  and  with  given  subject-matter.  It  should  help 
assure  a  uniformity  of  method  where  such  uniformity  is  desirable. 
For  instance,  if  the  Austrian  method  of  subtraction  is  used  by  some 
teachers,  it  should  be  used  uniformly  throughout  the  school  system. 
The  suggestions  on  appropriate  methods  should  follow  the  sugges- 
tions on  learning  exercises,  and  should  include  at  least  the  following 
items:  motivation  (including  games),  lesson  types,  oral  and  written 
work,  use  of  textbooks,  testing  and  remedial  instruction  (including 
standardized  and  informal  tests  and  diagnosis),  practice  tests, 
adaptation  of  instruction  to  individual  differences,  and  supervision  of 
study. 


22A  word  of  warning  is  not  amiss  here.  The  adaptation  of  learning  exercises  to 
local  conditions  should  not  be  carried  to  an  extreme.  For  a  good  example  of  what 
not  to  do.  see: 

Brown,  Joseph  C.  and  Coffmax,  Lotus  D.  How  to  Teach  Arithmetic. 
Chicago:    Row,  Peterson  and  Company,  1914,  p.  72. 

^For  a  digest  of  the  results  of  scientific  investigations  relative  to  methods 
of  instruction  in  arithmetic,  see: 

Monroe,  Walter  S.  ''Principles  of  method  in  teaching  arithmetic,  as  derived 
from  scientific  investigation."  Eighteenth  Yearbook  of  the  National  Society  for  the 
Study  of  Education.  Bloomington,  Illinois:  Public  School  Publishing  Company, 
1919,  p.  78-95. 


[16] 


Methods  of  instruction:  motivation.24  The  chief  assistance  on 
motivation  which  the  course  of  study  in  arithmetic  can  give  the 
teacher  is  in  suggesting  suitable  types  of  games,  field  trips,  and  sup- 
plementary projects.  These  are  learning  exercises  with  a  large 
motivating  element.  It  would  not  be  out  of  place  to  discuss  games 
and  field  trips  under  learning  exercises  rather  than  under  motivation, 
but  in  either  place  the  suggestions  should  be  of  a  concrete  rather 
than  of  a  theoretical  nature.  Specific  games  and  projects  should  be 
given.  This  is  another  excellent  opportunity  to  take  advantage  of 
local  conditions. 

The  course  of  study  should  make  clear  the  "motivating 
elements"25  that  are  effective  in  the  different  grades.  Curiosity, 
pleasure  in  successful  work,  manipulation,  and  many  other  bases  of 
motivation  are  powerful  throughout  life,  but  interests  as  motivating 
elements  change  greatly  from  grade  to  grade.  Means  of  motivation 
that  are  successful  in  the  second  grade  may  not  be  so  in  the  fourth, 
fifth  and  sixth  grades.  Those  motivating  factors  that  should  be 
avoided  and  those  of  which  most  advantage  should  be  taken  in  par- 
ticular grades  also  should  be  considered.  Some  teachers  have  a 
tendency  to  "over  motivate"  or  to  motivate  merely  for  the  sake  of 
motivation.  The  course  of  study  can  assist  in  overcoming  such  a 
habit  by  pointing  out  the  ill  effects  of  certain  practices  such  as  the 
use  of  distracting  games. 

Methods  of  instruction:  lesson  types.  In  arithmetic  there  are 
different  types  of  lessons.  The  three  outstanding  types  are  the 
development  lesson,26  the  drill  lesson,  and  the  lesson  that  involves 
plays  and  games.  The  course  of  study  should  make  a  distinction 
between  these  three  types  and  may  well  give  an  outline  of  an 
illustrative  lesson  for  each.  The  characteristics  of  the  good  drill 
lesson  should  be  pointed  out  especially,  for  there  has  probably  been 


24The  term  "motivation"  is  used  here  as  denned  by  Lennes  (p.  125) :  "Motiva- 
tion in  elementary  work  consists  in  so  combining  those  activities  which  are  desired 
of  the  child  with  other  activities  in  which  he  is  spontaneously  and  directly  interested 
that  the  combination  of  activities  will  to  him  be  interesting  and  attractive." 

For  an  excellent  discussion  of  motivation  and  fruitful  suggestions,  see: 

Lennes,  N.  J.  The  Teaching  of  Arithmetic.  New  York:  The  Macmillan 
Company,  1923,  p.  119-67. 

25The  term  "motivating  elements"  is  used  to  include  such  factors  as  instincts, 
acquired  interests  and  many  others  which  are  appealed  to  for  motivation. 

20For  illustrative  lesson  outlines,  see: 

"Arithmetic — elementary  course  of  study."  Trenton,  New  Jersey:  Board  of 
Education,   1923.      96  p. 

[17] 


too  much  of  a  tendency  to  waste  time  in  inefficient  drill  or  to  fail  to 
give  enough  drill.27  The  lesson  that  involves  plays  and  games  will 
need  little  more  than  mentioning  at  this  point  in  the  course  of  study 
because  of  the  discussion  elsewhere  under  either  motivation  or 
learning  exercises. 

Methods  of  instruction:  oral  and  written  work.  The  type, 
amount,  and  proportion  of  oral  and  written  work  in  arithmetic  vary 
from  grade  to  grade.  There  are  also  variations  made  in  the  time 
of  doing  the  written  work;  that  is,  in  recitation  time,  in  school  but 
outside  of  recitation  time,  or  at  home.  For  the  attainment  of  some 
objectives,  oral  work  is  the  more  effective,  for  others,  written  work  is 
more  successful.  Often  a  proper  proportion  of  each  is  desirable. 
Some  written  work  can  be  done  as  well  at  home  as  at  school;  some, 
especially  when  the  pupils  are  not  thoroughly  familiar  with  the 
process  involved,  should  be  done  only  under  the  supervision  of  the 
teacher.  On  all  such  questions,  the  course  of  study  should  give 
assistance  to  the  teacher. 

Methods  of  instruction:  use  of  textbooks.  The  textbook  in 
arithmetic  is  probably  more  used  and  more  slavishly  followed  than 
the  text  in  any  other  subject.  It  should  be  considered  a  guide  and 
an  economical  source  of  organized  subject-matter  and  learning 
exercises,  but  it  must  be  kept  subordinate  to  the  specific  objectives  of 
the  curriculum.  There  is  need  for  omitting  some  parts  as  well  as  for 
supplementing  other  parts.  The  course  of  study  should  point  out  these 
facts,  and  especially  should  show  from  grade  to  grade  the  specific 
use  that  should  be  made  of  the  textbook.  Undoubtedly  its  use  varies 
greatly;  in  the  first  grade  it  probably  should  not  be  introduced,  later 
it  becomes  an  important  instructional  instrument.  As  progress  is 
made  through  the  grades,  the  textbook  becomes  less  and  less  based 
upon  the  pupils'  experiences  and  gives  more  and  more  information 
other  than  arithmetical  knowledge.28 


27For  a  resume  of  the  studies  which  have  been  made  relative  to  drill  in  arith- 
metic, see: 

Wilson,  Guy  M.  "Arithmetic."  Third  Yearbook  of  the  Department  of 
Superintendence.  Washington:  Department  of  Superintendence  of  the  National 
Education  Association,  1925,  p.  63-91. 

2SFor  a  discussion  of  the  introduction  of  subject-matter  other  than  that  which 
is  purely  arithmetical,  see: 

Lennes,  X.  J.  The  Teaching  of  Arithmetic.  New  York:  The  Macmillan 
Company,  1923,  p.  171-90. 

[18] 


Methods  of  instruction:  use  of  tests.29  There  have  been  pub- 
lished a  large  number  of  standardized  tests  on  the  various  phases  of 
arithmetic.  They  may  be  used  by  the  classroom  teacher  for  the  fol- 
lowing purposes:  (1)  promotion  and  classification  of  pupils,  (2) 
diagnosis  of  pupils'  difficulties  in  order  to  provide  remedial  instruc- 
tion, and  (3)  evaluation  of  teaching  efficiency.  The  course  of  study 
should  not  attempt  to  be  a  treatise  on  the  use  of  standardized  tests, 
but  should  instruct  the  teacher  regarding  available  tests,  occasions 
for  their  administration,  and  the  uses  to  be  made  of  the  results.30 
From  the  standpoint  of  the  teacher,  remedial  instruction  is  the  most 
important  use  that  can  be  made  of  the  results  of  tests.  The  course  of 
study  should  point  out  the  types  of  difficulties  that  are  met  most 
often  and  some  of  the  means  of  overcoming  them.31  The  need  for  a 
great  deal  of  testing  besides  that  done  by  standardized  tests  should 
be  pointed  out  also.  Types  of  tests,  such  as  the  traditional  and  the 
new  examination,  and  the  uses  to  be  made  of  the  results  should  be 
discussed.32 

Methods  of  instruction:  practice  tests.33  A  number  of  lists  of 
exercises,  usually  called  practice  tests,  have  been  carefully  planned  so 
as  to  provide  the  drill  needed  to  engender  certain  specified  degrees 
of  ability.    If  such  practice  tests   are  considered  desirable   for  the 


29For  a  list  of  available  tests  and  a  discussion  of  their  purposes,  see: 

Doherty,  Margaret  and  MacLatchy,  Josephine.  "Bibliography  of  educa- 
tional and  psychological  tests  and  measurements."  U.  S.  Bureau  of  Education 
Bulletin,  1923,  No.  55.   Washington:  Government  Printing  Office,  1924.    233  p. 

Odell,  Charles  W.  "Educational  tests  for  use  in  elementary  schools,  revised." 
University  of  Illinois  Bulletin,  Vol.  22,  No.  16.  Bureau  of  Educational  Research 
Circular  No.  33.   Urbana:    University  of  Illinois,  1924.    22  p. 

30For  an  adequate  discussion  of  testing  in  arithmetic,  see: 

Monroe,  Walter  Scott,  DeVoss,  James  Clarence,  and  Kelly,  Frederick 
James.  Educational  Tests  and  Measurements.  (Revised  and  Enlarged.)  Boston: 
Houghton  Mifflin  Company,  1924,  p.  19-93. 

31The  most  recent  and  probably  the  best  discussion  of  remedial  instruction  in 
arithmetic  is: 

Osburn,  Worth  J.  Corrective  Arithmetic.  Boston:  Houghton  Mifflin  Com- 
pany, 1924.    182  p. 

32An  excellent  discussion  of  written  examinations  and  the  technique  of  informal 
testing  is  found  in: 

Monroe,  Walter  S.  and  Souders,  Lloyd  B.  "The  present  status  of  written 
examinations  and  suggestions  for  their  improvement."  University  of  Illinois  Bulletin, 
Vol.  21,  No.  13.  Bureau  of  Educational  Research  Bulletin  No.  17.  Urbana:  Univer- 
sity of  Illinois,  1923.   77  p. 

33The  bibliographies  just  referred  to  under  the  use  of  tests  include  practice  tests. 

[19] 


particular  school  system,  the  course  of  study  should  specify  those  to 
be  selected  and  make  provisions  for  their  systematic  use.  No  elabo- 
rate discussion  is  necessary  since  ample  directions  are  provided  by 
the  publishers,  but  the  teachers  should  familiarize  themselves  with 
these  instructions. 

Methods  of  instruction:  adaptation  of  instruction  to  individual 
differences.34  There  are  four  outstanding  types  of  provisions  for 
individual  differences  which  may  be  made:  first,  modifications  of 
objectives;  second,  modifications  of  topics;  third,  variations  in  types 
of  learning  exercises;  and,  fourth,  modifications  of  methods  of 
instruction.  The  modifications  of  objectives  and  topics  should  be  in- 
dicated in  the  first  outlines  of  specifications  given  in  the  introductory 
section  of  the  course  of  study  and  should  be  repeated  when  the  speci- 
fications are  made  for  each  grade.  Variations  in  learning  exercises 
and  modifications  of  methods  of  instruction  should  be  discussed  in 
the  portions  of  the  course  of  study  dealing  with  the  particular  grades. 
It  is  more  effective  to  group  these  suggestions  under  each  grade 
division  than  to  separate  them  into  distinct  topics. 

Methods  of  instruction:  directing  study.  The  most  important 
function  of  the  teacher  is  to  direct  the  learner  in  his  doing  of  learning 
exercises.  Much  of  the  activity  of  the  learner  must  be  carried  on  in 
so-called  study  periods,  whether  these  be  a  portion  of  the  recitation 
period  or  entirely  separate.  The  teacher's  success  depends  mainly 
upon  her  ability  to  direct  the  pupils  in  their  study.  She  must  be  able 
to  give  effective  directions  for  work,  to  know  when  to  give  assistance, 
and  to  decide  the  kind  of  assistance  which  will  be  most  helpful.  The 
course  of  study  cannot  give  the  teacher  all  of  this  ability,  but  it  can 
suggest  the  type  of  activity  on  her  part  which  is  most  likely  to  be 
effective.  Any  discussion  of  supervised  study  is  so  closely  related  to 
all  other  phases  of  instruction  that  ordinarily  there  should  be  a  gen- 
eral permeation  of  it  throughout  other  topics,  such  as  the  use  of  text- 
books, oral  and  written  work,  and  the  use  of  practice  tests.  If  in  a 
school  system  there  is  a  special  scheme  of  supervised  study,  such  as 
the  Batavia  Plan,  there  is  need  to  give  specific  directions  regarding 
the  particular  form  of  supervised  study  so  that  the  instruction  in 
arithmetic  may  fit  in  with  the  general  scheme. 


340ne  of  the  better  courses  of  study  In  its  provisions  for  individual  differences 
is  the  Long  Beach,  California,  course  of  study  in  arithmetic. 

[20] 


Methods  of  instruction:  summary.  Although  many  topics  (mo- 
tivation, lesson  types,  oral  and  written  work,  use  of  textbooks,  testing 
and  remedial  instruction,  practice  tests,  adaptation  of  instruction  to 
individual  differences  and  supervision  of  study)  have  been  discussed 
under  the  general  subject  of  methods  of  instruction,  all  of  these 
should  not  be  treated  in  great  detail  in  the  outlines  for  each  grade. 
Some  of  the  topics  will  need  to  be  dealt  with  much  more  extensively 
in  certain  grades  than  in  others.  The  manner  of  presenting  these 
various  topics  should  be  to  fuse  them  together  rather  than  to  force 
too  many  arbitrary  lines  of  distinction.  For  instance,  the  treatment 
of  lesson  types  and  of  oral  and  written  work  readily  fuse  into  each 
other,  yet  there  need  be  no  confusion  about  either.  If  the  reader 
will  again  refer  to  the  outline  of  a  course  of  study  in  arithmetic, 
pages  5-6,  this  fact  may  be  somewhat  more  evident. 


[21] 


APPENDIX 

COURSE  OF  STUDY  IN  ARITHMETIC 
GRADE  IV35 

The  textbook.  The  textbook  to  be  used  is  Drushel,  Noonan,  and 
Withers,  Arithmetic  Essentials — Book  One,36  p.  161-304. 

Practice  tests.  This  course  of  study  does  not  require  the  use  of 
practice  tests,  but,  if  they  are  desired  by  the  school,  it  makes  pro- 
visions for  the  Courtis  Standard  Practice  Tests.37 

Standardized  achievement  tests.  It  is  not  necessary  that  stand- 
ardized achievement  tests  be  used,  as  standards  of  achievement  are 
given  later  under  "objectives  and  standards  of  attainment."  However, 
if  the  use  of  such  tests  is  considered  desirable,  provision  is  made  for 
the  following:  Monroe's  General  Survey  Scales  in  Arithmetic,  Scale 
I,  forms  1  and  2;  Monroe's  Diagnostic  Tests  in  Arithmetic,  Parts  I 
and  II;  and  Monroe's  Standardized  Reasoning  Tests  in  Arithmetic.38 

I.  SPECIFICATIONS 

Objectives  and  standards  of  attainment.  The  following  objec- 
tives should  be  attained  before  pupils  are  promoted  to  the  fifth  grade. 
In  some  cases  the  degree  of  attainment  in  each  ability  is  not  stated 
and  must  be  left  to  the  teacher's  judgment.  But  for  the  fundamental 
operations  of  addition,  subtraction,  multiplication,  and  division,  the 
degrees  of  attainment  are  stated  and  should  be  adhered  to  strictly. 

The  time  allowances  in  parentheses  are  intended  for  the  slower 
pupils,  who  may  be  promoted  if  they  can  perform  the  operations 
accurately  within  the  somewhat  longer  time  limits.  Normal  and 
bright  pupils  should  be  held  to  the  shorter  time  limits.   They  should 


35As  an  example  of  an  arithmetic  course  of  study  constructed  in  accordance 
with  the  principles  set  forth  in  this  circular,  the  writer  has  prepared  this  portion  for 
the  fourth  grade.  It  is  not  intended  to  be  perfect,  but  it  does  furnish  a  fair  illustra- 
tion of  what  such  a  course  of  study  is  like.  It  is  assumed  that  there  are  similar 
sections  for  preceding  and  succeeding  grades,  and  an  introductory  section  as 
described  on  pages  7-9  of  this  circular. 

30Drushel,  J.  Andrew,  Noonan,  Margaret  E..  and  Withers,  John  W. 
Arithmetic  Essentials — Book  One.   Chicago:  Lyons  and  Carnahan,  1921.    304  p. 

37Published  by  the  World  Book  Company,  Yonkers-on-Hudson,  New  York, 
and  Chicago,  Illinois. 

38These  tests  are  published  by  the  Public  School  Publishing  Company,  Bloom- 
ington,  Illinois. 

[22] 


not  be  drilled  to  exceed  these  degrees  of  attainment,  but  they  should 
be  excused  from  further  drill  when  they  have  reached  these  standards 
and  for  so  long  a  time  as  they  maintain  them. 

Those  objectives  that  are  starred  (*)  may  be  omitted  for  the 
slower  pupils  without  hampering  them  materially  in  future  grades 
and  need  not  be  held  as  prerequisites  for  promotion. 

The  teacher  should  see  the  objectives  for  this  grade  in  their 
relation  to  all  the  objectives  for  elementary-school  arithmetic.  A 
tabulation  of  these  is  presented  in  the  introductory  section. 

Integers: 

1.  Ability  to  read  numbers  as  large  as  999,999,999. 

2.  Ability  to  write  numbers  as  large  as  999,999,999. 

*3.  Ability  to  read  Roman  numerals  through  C,  and  to  know   the  value  of 

D  and  M. 

*4.  Ability  to  write  Roman  numerals  through  C,  also  using  D  and  M. 

*5.  Ability  to  tell  time  on  a  clock  which  has  Roman  numerals. 

*6.  Ability  to  tell  the  days  of  the  week  and  month  on  a  calendar. 

*7.  Ability  to  read  a  Fahrenheit  thermometer. 

8.  Ability  to  recall  the  multiplication  tables  of  10's,  ll's,  and  12's  through  12. 

9.  Ability  to  do  the  following  types  of  addition  examples  at  the  following 
rates: 


Example 

Number 

Time 

Example 

Number 

Time 

4 

7 

7 

17 

1  min. 

6 

2 

OK  ") 

6 
5 
0 

927 

379 

5 

4 

4  min. 

756 

1 

(6     "   ) 

837 

6 

8  min. 

8 

924 

(12     "    ) 

7 

110 

3 

854 

3 

965 

1 

344 

2 

10.  Ability  to  do  the  following  types  of  subtraction  examples  at  the  following 
rates: 

Example       Number  Time         Example       Number  Time 

107795491  7  4  min.  37  6  1  min. 

77197029  (6     "   )  _5_  (IK    "  ) 

739  3  1  min. 

367  (IK    "   ) 

11.  Ability  to  do  the  following  types  of  multiplication  examples  at  the  fol- 
lowing rates: 

Example       Number  Time         Example       Number  Time 

(>S12  3  1  min.  8246  6  6  min. 

6  (IK     ")  29  (9     "    ) 


[23] 


12.  Ability  to  do  the  following  types  of  division  examples  at  the  following 
rates: 

Example       Number  Time         Example       Number  Time 


Example 

Number 

Time 

3/5  +  1/5  = 

14 

1  min. 

8)3840  1  1  min.  25)6775  4  8  min. 

(IK     ")  (12     "     ) 

Common  fractions: 

1.  Ability  to  find  1/6  and  1/8  of  single  things. 

2.  Ability  to  find  1/2,  1/3,  2/3,  1/4,  3/4,  1/5,  2/5,  3/5,  4/5,  1/6,  S/6,  1/8, 
3/8,  5/8,  1/9,  1/10,  3/10,  7/10,  and  9/10  of  groups  of  things.  (Without  remainders 
for  slower  pupils.) 

3.  Ability  to  find  1/11  of  multiples  of  1 1  through  132. 

4.  Ability  to  find  1/12  of  multiples  of  12  through  144. 

5.  Ability  to  do  the  following  types  of  addition  and  substraction  examples 
withjfractions  at  the  following  rates: 

Example       Number  Time 

6/9-4/9=        14  1  min. 

)  OK"      ) 

Decimals: 

1.  Ability  to  read  U.S.  money  expressed  decimally  as  large  as  $999,999,999.99. 

2.  Ability  to  write  U.  S.  money  expressed  decimally  as  large  as 
$999,999,999.99. 

3.  Ability  to  multiply  U.  S.  money  expressed  decimally  as  large  as 
$999,999.99  by  a  two-place  multiplier. 

4.  Ability  to  divide  U.  S.  money  expressed  decimally  as  large  as  $999,999.99 
by  a  two-place  divisor. 

Denominate  numbers: 

1.  Ability  to  recall  the  following  tables  as  given  in  their  limited  form  on  page 

304  of  the  textbook: 

Money  Length  Area  Liquid  measure 

Time  Weight  Dry  measure 

Practical  measurements: 

1.  Ability  to  make  change  with  amounts  not  larger  than  one  dollar. 

2.  Ability  to  measure  accessible  rectangular  surfaces  within  limits  of  one 
acre. 

3.  Ability  to  estimate  reasonably  well  rectangular  surfaces  within  limits  of 
one  square  rod. 

4.  Ability  to  weigh  objects  within  limits  of  one  hundred  pounds. 

5.  Ability  to  estimate  reasonably  well  the  weight  of  objects  within  limits  of 
fifty  pounds. 

6.  Ability  to  measure  liquids  within  limits  of  five  gallons. 

7.  Ability  to  estimate  reasonably  well  quantity  of  liquids  within  limits  of 
ten  gallons. 

8.  Ability  to  measure  grains  and  the  like  within  limits  of  one  bushel. 

9.  Ability  to  estimate  reasonably  well  quantity  of  grains  and  the  like  within 
limits  of  five  bushels. 

Problems: 

1.  Ability  to  solve  one-step  problems  that  involve  the  subject-matter 
(denominate  numbers,  decimals,  etc.)  of  this  grade.      For  example: 

A  boy  mows  lawns  for  twenty-five  cents  an  hour.     It  took  him  five 
hours  to  mow  a  large  lawn.     How  much  did  he  make? 

[24] 


2.  Ability  to  solve  two-step  problems  that  involve  making  change  within 
limits  of  $100.00.     For  example: 

A  boy  bought  a  pencil  tablet  for  ten  cents  and  a  pencil  for  three  cents. 

He  gave  the  clerk  twenty-five  cents.     How  much  change  should  he  receive? 

Note — Two-step  problems,  other  than  those  involving  the  making  of 
change,  and  problems  involving  more  than  two  steps  in  making  change,  should 
not  be  attempted  in  this  grade. 

Nomenclature  and  symbols: 

1.  Ability  to  use  the  following  terms: 

A.  M.  multiplicand  plus 

P.  M.  multiplier  less 

addend  product  multiplied  by 

sum  long  division  divided  by 

minuend  dividend  integer 

subtrahend  divisor  common  fraction 

difference  quotient  Arabic  numeral 

remainder  Roman  numeral 

Generalized  habits:     (To  be  striven  for  in  all  grades,  but  especially  emphasized  in 
the  fourth.) 

1.  Habit  of  checking  answers. 

2.  Habit  of  thinking  through  the  solution  of  a  problem  before  doing  the 
computation. 

3.  Habit  of  working  with  initiative  and  independence. 

Attitudes:     (To  be  striven  for  in  all  grades,  but  especially  emphasized  in  the  fourth.) 

1.  Pride  in  one's  ability  to  use  numbers. 

2.  Pride  in  one's  ability  to  attain  standard  achievement. 

Outline  of  subject-matter.  The  subject-matter  outlined  below  is  essentially 
that  presented  in  the  textbook  but  is  presented  here  in  a  logical  outline  in  order 
that  the  teacher  may  see  the  work  of  this  grade  as  a  whole  and  in  proper  relation 
to  the  objectives  to  be  attained.  She  should  also  see  the  work  of  this  grade  in 
its  proper  relation  to  the  entire  arithmetic  curriculum.  This  is  presented  in 
tabular  form  in  the  introductory  section. 

The  subject-matter  that  is  starred  (*)  is  not  essential  to  future  progress 
and  may  be  omitted  by  the  slower  pupils.  There  are  relatively  few  subject- 
matter  provisions  for  individual  differences  in  this  grade.  Since  arithmetic  is  a 
tool  subject,  there  should  be  comparatively  little  difference  in  the  subject- 
matter  of  this  grade  for  pupils  of  various  abilities.  The  chief  provisions  should 
be  excusing  from  additional  work  when  standards  of  attainment  and  objectives 
have  been  achieved  and  making  modifications  of  method.  These  will  be  discussed 
in  a  later  section  on  methods  of  teaching,  pages  32-35. 

Integers: 

1.  Arabic  notation  and  numeration  through  hundred  millions. 
*2.  Roman  numbers  through  C.     Also  D  and  M. 

3.  Addition. 

4.  Subtraction. 

5.  Multiplication — tables  of  10's,   ll's,   and   12's  through   12. 

6.  Short  division. 

7.  Long  division  with  one  and  two-place  divisors  and  three  and  four-place 
dividends. 


[25] 


Common  fractions: 

1.  1/2,  1/3,  2/3,  1/4,  3/4,  1/5,  2/5,  3/5,  4/5,  1/6,  5/6,  1/8,  3/8,  5/8,  1/9, 
1/10,  3/10,  7/10,  9/10. 

2.  Addition  of  fractions  of  same  denominator. 

3.  Subtraction  effractions  of  same  denominator. 

Decimals: 

1.  U.  S.  money  to  $999,999,999.99. 

2.  Multiplication  of  U.  S.  money  as  large  as  $999,999.99  by  one  and  two- 
place  multipliers. 

3.  Division  of  U.  S.  money  as  large  as  $999,999.99  by  one  and  two-place 
divisors. 

Denominate  numbers: 

1.  Time: 

Second,  minute,  hour,  day,  week,  year,  leap  year,  century. 

2.  Money: 

Cent,  dime,  quarter,  half  dollar,  dollar. 

3.  Length: 

Inch,  foot,  yard,  rod,  mile. 

4.  Weight: 

Ounce,  pound,  ton. 

5.  Area. 

Square  inch,  square  foot,  square  yard,  square  rod. 

6.  Liquid  measure: 

Pint,    quart,   gallon. 

7.  Dry  measure: 

Quart,  peck,  bushel. 

II.   DIRECTIONS  RELATIVE  TO  INSTRUCTION 
1.    Learning  Exercises 

Textbook  largely  composed  of  learning  exercises.  The  arith- 
metic textbook  is  composed  chiefly  of  learning  exercises:  problems, 
examples,  drill  exercises,  and  so  forth.  The  task  of  the  arithmetic 
teacher  with  respect  to  learning  exercises  is  light  as  compared  with 
that  of  teachers  of  other  subjects,  such  as  history  where  compar- 
atively few  such  exercises  are  given  in  the  text.  The  principal  duties 
of  the  arithmetic  teacher  in  this  respect  are  first,  to  select  the  appro- 
priate exercises  from  those  given  in  the  textbook;  and  second,  to  find 
or  formulate  supplementary  learning  exercises  when  those  given  are 
not  sufficient  to  achieve  a  given  objective,  when  they  are  inappro- 
priate, when  the  teacher  sees  an  opportunity  to  take  advantage  of  her 
pupils'  interest  or  of  peculiar  local  conditions,  or  when  some  other 
need  or  opportunity  arises. 

Additional  learning  exercises  needed.  There  are  a  few  places  in 
Drushel,  Noonan,  and  Withers'  book  which  are  particularly  weak, 
and   which   should   be   supplemented   either  by  the   use  of   supple- 

[26] 


mentary  textbooks  or  by  exercises  devised  by  the  teacher.  One  such 
place  is  on  page  166-67  where  the  topic  of  Roman  numerals  is  taken 
up.  Those  pupils  who  are  taught  to  read,  write,  and  use  Roman 
numerals  must  be  given  more  practice  than  is  provided  here  if  they 
are  to  attain  any  real  proficiency.  The  topic  should  be  developed 
much  more  gradually.  The  teacher  should  have  the  pupils  write 
numbers  in  succession  from  1  to  100,  giving  both  the  Arabic  and  the 
Roman.  A  bird's  eye  view  of  all  of  the  letters  used  may  be  given  at 
first,  but  no  emphasis  should  be  placed  upon  the  individual  letters 
until  they  are  needed.  Thus,  the  meaning  of  L  should  not  be  stressed 
until  the  pupils  are  ready  to  write  fifty.  After  the  pupils  are  able  to 
write,  to  XII,  they  may  be  taught  to  tell  time  on  a  watch  or  clock 
which  has  Roman  numerals.  They  should  also  read  the  numbers  of 
chapters  in  books,  the  numbers  of  volumes  of  papers  and  magazines; 
and  if  interest  is  sufficiently  aroused,  they  may  learn  to  read  dates 
on  corner  stones  of  buildings. 

There  is  little  reason  "for  teaching  any  Roman  numerals  except 
those  from  one  to  twenty,  fifty,  one  hundred,  five  hundred,  and  one 
thousand;"39  and  the  more  gifted  pupils  are  probably  the  only  ones 
who  should  be  taught  these.  But  Roman  numerals,  when  taught  at 
all,  should  be  so  thoroughly  taught  that  there  is  real  mastery. 
"Smattering"  is  not  justifiable.  Frequent  reviews  and  repetitions 
throughout  the  year  will  be  necessary. 

The  objective  of  reading  and  writing  Arabic  numbers  as  large 
as  999,999,999  is  not  provided  for  in  the  exercises  of  the  text.  The 
work  on  page  189  should  be  extended  to  include  such  numbers. 
Pupils  of  the  fourth  grade  usually  like  to  read,  write,  and  "say" 
large  numbers.  Little  contests  similar  to  "spell-downs"  can  be  used 
to  help  teach  these  large  numbers,  and  newspaper  articles  containing 
large  numbers  can  be  brought  for  the  pupils  to  read.  In  all  teaching 
of  numbers  there  should  be  a  minimum  amount  of  attention,  prob- 
ably none,  given  to  the  reading  of  units,  tens,  hundreds,  thousands, 
and  so  forth;  but  rather  the  pupils  should  be  taught  at  first  merely 
to  read  the  numbers  as  they  are  ordinarily  read:  54  (fifty-four),  123 
(one  hundred  twenty-three),  and  so  forth.  "Beginning  with  the 
thousands  pupils  should  be  taught  to  read  by  the  'period  method,' 


39Brown,   Joseph   C.   and   Coffman,   Lotus    D.     How   to   Teach   Arithmetic. 
Chicago:    Row,  Peterson  and  Company,  1914,  p.  147. 


[27] 


designating  each  group  of  three  number  places  as  shown  in  the  fol- 
lowing scheme: 

billions  millions  thousands  ones 

000  000  000  000 

12  638 

146  921  345 

9  000  000  000 

"A  number  in  any  period  is  read  as  is  any  one,  two,  or  three 
place  number  and  is  followed  by  the  name  of  the  period  in  which  it 
is  placed.  The  name  of  the  'ones'  period  is  always  omitted  in  read- 
ing numbers.  Pupils  should  be  trained  to  avoid  the  use  of  'and'  in 
reading  whole  numbers."40 

The  teaching  of  long  division  is  one  of  the  most  difficult  tasks  of 
the  fourth  grade.  The  learning  exercises  provided  in  the  textbook  are 
adequate  and  well-graded  for  most  children,  so  that  there  will  be 
relatively  little  need  to  devise  supplementary  ones.  But  in  order  to 
teach  most  effectively,  the  teacher  should  be  sure  that  she  thoroughly 
understands  the  difficulties  to  be  met  and  the  types  of  exercises  that 
are  best  suited  to  meet  these  difficulties.  Especially  should  care  be 
taken  that  the  pupils  do  not  establish  bad  habits  of  work  or  in- 
hibitions due  to  beginning  with  exercises  that  involve  too  many  or 
too  great  difficulties  or  to  moving  forward  too  rapidly.  Osburn41  in 
his  Corrective  Arithmetic  analyzes  the  situation  well.  Space  does 
not  permit  a  repetition  here,  but  the  teacher  should  familiarize  her- 
self with  this  discussion. 

With  the  exception  of  long  division,  the  teaching  of  fractions  is 
the  most  difficult  topic  in  the  fourth  grade.  Here  again  Osburn's 
book  offers  some  assistance.  The  teacher  should  read  pages  47,  50, 
52,  and  54.  The  work  in  fractions  is  left  until  the  end  of  the  year  and 
is  so  gradually  and  thoroughly  developed  that  there  is  no  need  for 
the  teacher  to  devise  additional  learning  exercises  for  most  pupils. 

Drushel,  Noonan,  and  Withers'  book  has  a  great  variety  of 
learning  exercises  which  in  their  general  way  are  closely  tied  up  with 
the  everyday  life  of  the  pupils.  But  often  they  lack  point  because 
they  are  phrased  in  broad  terms  such  as  "Some  boys  in  the  fourth 


^•'Arithmetic — course  of  study  for  grades  four,  five,  and  six."  Baltimore,  Mary- 
land:   Department  of  Education,  1924,  p.  24-25. 

"Osburn,  Worth  J.  Corrective  Arithmetic.  Boston:  Houghton  Mifflin  Com- 
pany, 1924,  p.  68-70,  173-77. 

[28] 


grade."42  Such  exercises  are  very  suggestive  and  can  often  be 
changed  by  the  teacher  or  used  as  models  for  other  exercises  that  will 
be  even  more  "alive"  to  the  pupils.  Whenever  making  such  changes 
or  additions,  the  teacher  should  remember  that  these  exercises  have 
been  carefully  arranged  and  graded  so  as  to  introduce  the  pupils  to 
new  processes  gradually  and  to  review  thoroughly  old  processes. 
Care  should  be  taken  not  to  violate  these  precautions.  One  of  the 
greatest  dangers  in  so-called  "practical"  problems  is  the  too  sudden 
or  too  rapid  introduction  of  new  difficulties.  Another  danger  is  the 
entire  neglect  of  some  phases  of  processes  which  must  be  given  atten- 
tion if  the  processes  are  to  be  fully  mastered.  If  the  teacher  avoids 
these  two  dangers  and  knows  what  purposes  she  means  to  accomplish 
by  exercises  that  she  may  devise  or  have  the  pupils  devise,  she  will 
often  accomplish  results  which  could  not  be  achieved  by  using  the 
book  exercises  alone. 

Some  additional  exercises  that  may  be  worked  out,  often  in  the 
form  of  projects  by  the  brighter  pupils,  are  the  following: 

A  live  map.43 

Making  a  toy  shop.44 

How  I  may  save  and  help  my  parents  to  save. 

Operating  a  ticket  office. 

Keeping  a  general  merchandise  store. 

Raising  money  by  selling  garden  produce. 

Making  money  by  raising  rabbits. 
Accuracy,  speed  and  checking  answers.  In  all  exercises  accu- 
racy should  be  insisted  upon.  It  is  not  desirable  to  urge  accuracy  at 
the  sacrifice  of  speed.45  Rather,  accuracy  should  be  secured  by  means 
of  checking  answers  and  by  placing  emphasis  upon  speed.  The  fol- 
lowing checks46  may  be  introduced  in  the  fourth  grade. 


42Drushel,  J.  Andrew,  Noonan,  Margaret  E.,  and  Withers,  John  W. 
Arithmetic  Essentials — Book  One.    Chicago:  Lyons  and  Carnahan,  1921,  p.  161. 

^Twentieth  Yearbook  of  the  National  Society  for  the  Study  of  Education. 
Bloomington,  Illinois:    Public  School  Publishing  Company,  1921,  p.  39. 

"Ibid.,  p.  57. 

45Monroe,  Walter  S.  "Principles  of  method  in  teaching  arithmetic,  as 
derived  from  scientific  investigation."  Eighteenth  Yearbook  of  the  National  Society 
for  the  Study  of  Education,  Part  II.  Bloomington,  Illinois:  Public  School  Publish- 
ing Company,  1919,  p.  89. 

""Quoted  from: 

"Arithmetic — course  of  study  for  grades  four,  five,  and  six."  Baltimore,  Mary- 
land:   Board  of  Education,   1924,  p.  30-32. 

[29] 


1.  Approximating  results: 

Approximation  of  results,  i.e.,  estimating  the  results  through  a  reasonable  com- 
parison of  the  facts  given  and  required,  is  a  most  valuable  check  in  the  solution 
of  problems,  immediately  showing  the  absurdity  of  a  grossly  incorrect  answer. 

For  example,  in  the  problem,  "One  book  cost  $.75.  Find  the  cost  of  12  books," 
pupils  should  be  taught  to  approximate  the  answer  as  follows: 

12   books  at  $1.00  each  will  cost  $12.00.    $.75    is   less  than  one   dollar, 

therefore,  12  books  at  $.75  will  cost  less  than  $12.00. 

This  approximation  would  immediately  show  the  absurdity  of  such  a  result  as 
$900.00,  which  was  the  answer  actually  given  by  a  pupil  with  no  appreciation  of 
the  fact  that  the  answer  was  not  reasonable. 

2.  Check  for  addition: 

The  best  check  for  addition  is  to  add  each  column  from  the  bottom  up, 
writing  the  sum  at  the  right  of  the  column;  then  to  add  the  same  column  from  the 
top  down,  checking  the  sum  if  correct.  Write  the  right  hand  digit  of  the  sum 
beneath  the  column  added  and  add  the  other  digit  to  the  next  column. 

624  27  V 

239  18  V 

876  22  V 

548 


2287 
Reference:     Bailey,   Teaching  Arithmetic,   p.   68. 
3.  Check  for  subtraction: 

To   check   subtraction,    add    the   subtrahend    and   remainder.      The   result 
should  be  the  minuend. 

179807 
94926 


84881 
Without  re-writing  any  figures,  have  pupils  add  the  subtrahend  and  the 
remainder  as  follows: 

6  and  1  are  7,  2  and  8  are  10,  carry  1;  10  and  8  are  18;  carry  1;  5  and  4 
are  9;  9  and  8  are  17. 
4.  To  check  multiplication: 

a.   Multiply  the  multiplier  by  the  multiplicand. 

Original  example  Check 

728  35 

35  728 

3640  280 

2184  70 


25480 


245 


25480 
b.   Divide  the  product  by  either  of  the  factors;  the  result  should  be  the 
other  factor. 

728 

35)25480 
245 
98 
70 
280 
280 


[30] 


5.  Check  for  division: 

Multiply  the  quotient  by  the  divisor.     If  there  is  a  remainder  add  it  to  the 
product.     The  result  should  equal  the  dividend. 

Original  example  Check 

24  24 


:)7614 

312 

624 

48 

1374 

24 

1248 

72 

126 

7488 

126 

7614 

Note — The  quotient  should  not  be  used  as  a  multiplier  as  errors  occurring 
in  the  original  example  may  be  repeated  in  the  check. 

Short  cuts.    Speed  may  be  furthered  by  the  use  of  short  cuts 

that    are    real    time-saving   devices.     The    following    short   methods 

should  be  taught  in  the  fourth  grade.47 

1.  Addition: 

Addition  of  2  numbers  of  2  orders  may  be  taught  by  adding,  first,  the  tens 
digits  and  then,  the  units  digits  of  the  addends. 

For  example:  To  find  the  sum  of  38  and  43. 
Think  38  +  40  +  3.     Say  38,  78,  81. 

2.  Subtraction 

To  subtract  43  from  81,  reverse  the  process  used  for  addition. 
Think  81  minus  40  =  41;  41  minus  3  =  38. 
Say  81,41,  38. 

Reference:  Klapper,  The  Teaching  of  Arithmetic,  p.  166,  173. 

3.  Multiplication  by  10,  100,  and  1000. 

Children  should  be  taught  to  use  the  short  method  of  multiplying  by  powers 
of  10,  i.e.,  to  multiply  an  integer  by  10,  annex  one  zero;  to  multiply  by  100, 
annex  two  zeros;  to  multiply  by  1000,  annex  three  zeros. 

38  45  76 

10  100  1000 

380  4500  76000 

Reference:    Wentworth-Smith,  School  Arithmetics.    Book  One,  p.  132. 

4.  Division  by  10,  100,  and  1000. 

To  divide  an  integer  by  10,  100,  or  1000  cut  off  from  the  right  of  the  dividend 
as  many  figures  as  there  are  zeros  in  the  divisor. 

The  quotient  is  the  number  expressed  by  the  figures  at  the  left  of  the  line. 
The  remainder  is  the  number  expressed  by  the  figures  at  the  right. 
10)380  100)4579  1000)58371 

79_  371 

38  45  100  581000 


1|0)38|0  1|00)45|79  1|000)58|37: 


7Quoted  from: 

'Arithmetic — course  of  study  for  grades  four,  five,  and  six."    Baltimore,  Mary- 
land:   Board  of  Education,  1924,  p.  32-33. 


[31] 


2.    Methods  of  Instruction 

Motivation.  The  exercises  in  Drushel,  Noonan,  and  Witners' 
book  have  in  themselves  a  very  large  motivating  element,  that  is, 
they  interest  the  pupils  because  of  their  nature.  For  instance,  there 
are  colored  pictures,  other  children  are  represented  as  doing  things 
(things  that  children  like  to  do),  there  is  a  great  deal  of  variety  in 
the  kinds  of  things  about  which  problems  are  made,  there  are  things 
suggested  which  pupils  can  make  at  school  or  at  home,  and  there  are 
other  similar  motivating  elements.  But  the  teacher  should  not  forget 
that  there  are  many  bases  of  motivation  to  which  she  may  appeal 
and  of  which  she  should  take  advantage.  Among  those  that  are 
especially  good  in  the  fourth  grade  are  physical  activity,  desire  to 
construct  things,  personal  rivalry,  group  rivalry,  the  "I'll  show  you" 
attitude  toward  trying  tasks,  interest  in  the  familiar,  interest  in  the 
novel,  imitation,  desire  for  excellence,  self-esteem,  and  curiosity.  Of 
these,  the  four,  the  "I'll  show  you"  spirit,  group  rivalry,  desire  for 
excellence,  and  self-esteem,  should  be  given  particular  attention  be- 
cause, at  about  this  time  in  the  pupils'  lives,  such  attitudes  are 
developing  rapidly  and  should  be  directed  toward  desirable  ends. 

Games.  As  noted  in  the  preceding  paragraph,  the  learning  exer- 
cises, particularly  the  problems,  have  a  large  motivating  element  in 
themselves.  There  are  less  inherent  motivating  elements  in  the  drill 
exercises.  Because  of  this,  the  teacher  will  need  to  give  more  attention 
to  motivation  in  drill  exercises  than  in  problems.  Much  of  this  can  be 
done  through  games.  The  teacher  should  devise  or  select  suitable 
games,48  but  she  must  take  care  that  the  games  are  effective  as 
learning  exercises  and  not  merely  entertaining  or  distracting. 
Lennes49  lists  the  following  characteristics  that  should  be  possessed 
by  a  good  number  game: 

a.  It  should  keep  all  the  children  at  work  during  practically 
the  whole  period  devoted  to  the  game. 

b.  It  should  not  lead  to  noise  or  confusion  which  distracts 
the  attention  from  the  subject. 

c.  It  should  create  in  the  child  a  keen  desire  to  learn. 

d.  It  should  make  the  element  of  group  rivalry  stand  out 
sharply. 


48Some  of  the  best  suggestions  are  to  be  found  in  educational  periodicals  and  in 
books  on  the  teaching  of  arithmetic. 

49Lennes,  N.  J.  The  Teaching  of  Arithmetic.  New  York:  The  Macmillan 
Company,  1923,  p.  141-45. 

[32] 


e.  It  should  make  the  group  exert  pressure  on  the  individual 
o  do  his  best  work. 

f.  It  should  make  each  pupil  vitally  interested  in  the  work 
of  the  other  members  of  his  group. 

The  following  example  of  a  good  game  is  given:  A  class  is 
divided  into  opposing  teams,  and  perhaps  given  the  names  of  two 
colleges  or  of  two  athletic  teams  in  which  all  the  pupils  are  interested. 
All  are  provided  with  pencils  and  paper.  As  the  teacher  shows  cards, 
each  of  which  contains  two  numbers  to  be  added  (they  could  just  as 
well  be  for  multiplication,  subtraction,  or  division),  the  pupils  write 
down  the  sum.  After  a  certain  number  of  cards,  for  instance  ten, 
have  been  shown,  the  pupils  trade  papers  so  that  each  has  a  paper 
for  some  member  of  the  opposing  team.  Then  the  teacher  reads  the 
correct  answers  in  the  order  in  which  they  were  given,  and  the 
number  of  mistakes  for  each  team  is  quickly  found.  This  is  a  very 
simple  game,  but  it  possesses  all  the  good  elements  enumerated  on 
page  32,  especially  if  it  is  played  from  time  to  time  with  the  same 
teams  and  a  record  is  kept  of  the  scores.  The  game  may  be  varied 
in  several  ways,  such  as,  by  having  the  boys  and  girls  oppose  each 
other  or  by  selecting  captains  to  choose  teams. 

The  following  game  is  given  by  Lennes  as  an  example  of  a  bad 
game:  One  pupil  is  "it."  He  stands  before  the  class  and  says,  "I  am 
thinking  of  two  numbers  whose  sum  is  fifteen,"  or  any  other  number 
he  may  have  in  mind.  The  other  children  then  ask  in  turns,  "Is  it 
seven  and  eight  are  fifteen?"  "Is  it  ten  and  five?"  "Is  it  nine  and 
six?"  and  so  on  until  someone  guesses  the  combination  of  which  the 
first  pupil  is  thinking.  The  one  who  first  guesses  right  is  then  "it" 
and  the  process  is  started  all  over  again.  The  bad  elements  in  this 
game  are  apparent.  The  pupils  cannot  possibly  all  be  kept  busy  any 
considerable  part  of  the  time,  for  as  soon  as  a  pupil  has  thought  of 
a  combination,  all  he  has  to  do  is  to  keep  it  in  mind  until  his  turn 
comes  or  until  someone  else  gives  the  same  combination.  There  is 
likely  to  be  little  real  rivalry,  for  the  element  of  chance  is  too  prom- 
inent. Furthermore  the  rivalry  that  exists  is  individual,  and  the 
game  could  scarcely  be  so  modified  as  to  bring  about  group  rivalry. 

Evaluation  of  these  two  games  by  means  of  the  criteria  for 
judging  a  number  game,  page  32,  will  easily  bring  out  other  points  of 
excellence  and  of  weakness.  These  criteria  should  be  used  by 
teachers  whenever  their  pupils  play  games,  for  the  harm  resulting 


[33] 


from  a  poor  game  may  be  as  great  as  the  good  that  could  be  accom- 
plished by  a  well-chosen  game. 

Oral  and  written  work.  Little  needs  to  be  said  here  about  the 
proportion  of  oral  to  written  work,  for  adequate  provision  is  made 
in  the  textbook.  When  omissions  are  made  from  the  text,  care  should 
be  taken  not  to  underestimate  the  value  of  oral  work.  Rapid  written 
work  on  problems  does  not  imply  corresponding  speed  and  accuracy 
on  oral  ones.  Many  oral  exercises  and  problems  are  needed  if  skill  is 
expected  in  purely  mental  operations.  Furthermore,  oral  problems 
(simple  ones,  it  is  true)  are  assuming  a  larger  and  larger  place  in 
everyday  life,  so  that  training  in  them  is  appropriate  not  only  for 
obtaining  facility  in  mental  processes  but  also  for  meeting  life 
situations. 

Proper  use  of  the  textbook.  As  has  been  indicated  throughout 
the  preceding  pages,  the  textbook  is  the  chief  source  of  subject- 
matter  and  learning  exercises,  although  it  is  not  to  be  followed 
slavishly.  Several  places  have  been  pointed  out  as  in  need  of  sup- 
plementing. Each  teacher  will  find  other  places  that  need  supple- 
menting for  her  class  as  well  as  some  that  may  be  omitted  because 
of  the  special  aptitudes  of  her  pupils.  Most  of  these  minor  variations 
must  be  left  to  the  individual  teacher.  She  must  remember  that  the 
textbook  is  the  most  economical  source  of  subject-matter  and  of 
learning  exercises  which  she  has  available  and  that  variations  from 
it  are  justifiable  only  in  the  degree  to  which  they  aid  in  achieving 
more  efficiently  the  objectives  of  the  course  in  arithmetic. 

Use  of  practice  tests.  If  the  Courtis  Standard  Practice  Tests  are 
employed,  they  should  be  used  systematically  and  according  to  the 
accompanying  instructions.  Even  though  a  teacher  may  have  used 
the  tests  before,  she  should  read  again  the  instructions  and  sugges- 
tions; for  often  she  will  discover  that  she  has  previously  overlooked 
ways  in  which  she  could  use  the  tests  more  effectively  and  thus  secure 
better  results.  These  tests  furnish  only  a  limited  type  of  training 
and  test  only  particular  abilities.  Other  phases  of  arithmetic,  such  as 
problem  solving,  must  not  be  neglected. 

Provisions  for  individual  differences.  The  use  of  practice  tests 
enables  the  teacher  to  make  some  provisions  for  individual  differ- 
ences, such  as  excusing  from  further  drill  those  pupils  who  have 
achieved  the  standard  attainment.  Other  provisions  for  individual 
differences  are  made  in  the  statement  of  specifications,  pages  23-26. 

[34] 


A  few  objectives  and  topics  are  starred  because  they  are  intended 
only  for  the  brighter  pupils,  and  extended  time  limits  are  set  for  the 
slower  pupils  on  some  of  the  standards  of  attainment.  But  there  are 
several  provisions  for  individual  differences  which  more  properly 
come  under  method,  if  the  term  method  is  extended  to  include  learn- 
ing exercises  and  management.  For  example,  there  may  be  no  need 
for  the  brighter  pupils  to  do  the  practice  exercises  on  pages  192-93 ; 
for  if  they  have  attained  the  standard  abilities  in  the  processes  in- 
volved by  the  time  these  exercises  are  reached,  the  extra  practice 
would  be  undesirable.50  For  some  of  the  slower  pupils,  it  may  be 
necessary  to  devise  additional  exercises  in  order  that  they  may  attain 
the  desired  ability.  The  brighter  children  may  often  be  set  at  acquir- 
ing skill  after  having  a  principle  explained  to  them  while  the  slower 
ones  are  retained  for  work  preliminary  to  starting  a  principle.  Thus, 
in  learning  to  multiply  or  divide  by  multiples  of  ten,  the  general 
principles  of  adding  zeros  to  the  multiplicand  or  of  cutting  of!  a  cor- 
responding number  of  places  from  the  dividend  may  be  grasped 
immediately  by  some  pupils,  while  others  will  need  to  do  a  number 
of  preliminary  exercises  before  they  grasp  the  general  principle 
underlying  the  processes. 

Especial  attention  needs  to  be  given  to  discovering  and  appeal- 
ing to  the  interests  of  the  slower  pupils  in  order  that  their  interest  in 
arithmetic  may  be  aroused  and  maintained.  Ordinarily  the  bright 
pupils  like  whatever  they  are  asked  to  do,  but  they  are  in  danger  of 
being  bored  when  the  tasks  are  too  easy  or  when  there  is  too  much 
repetition. 

The  biggest  question  that  arises  in  providing  for  individual  dif- 
ferences is  more  often  one  of  management  than  of  method  or 
curriculum.  While  the  slower  pupils  are  still  working,  what  is  to  be 
done  with  those  who  have  completed  the  task?  Small  adjustments 
in  assignments  of  written  work  easily  care  for  minor  variations,  but 
large  differences  must  be  provided  for  in  some  other  way.  If  pupils 
are  excused  from  work  of  the  type  that  most  of  the  class  are  on,  they 
may  take  up  some  other  subject,  but  ordinarily  they  have  as  little 
need  of  additional  work  in  other  subjects  as  in  arithmetic.  An  oppor- 
tunity to  enrich  their  arithmetical  concepts  is  offered  to  the  teacher. 
Projects  such  as  those  suggested  for  additional  learning  exercises  on 


50When  pupils  are  excused  from  further  practice  on  a  given  process,  they 
should  be  tested  at  intervals  in  order  to  make  sure  that  they  are  maintaining  the 
standard  ability. 

[35] 


page  29  allow  the  brighter  pupils  to  exercise  2nd  develop  further  the 
initiative  they  already  possess. 

All  provisions  for  individual  differences  should  be  based  upon 
definitely  established  facts  in  so  far  as  possible.  This  means  that 
there  must  be  constant  testing.  Fortunately,  the  textbook  in  this 
grade  provides  a  number  of  tests  on  the  fundamental  operations 
which  may  be  used;  but  there  are  not  enough  tests,  such  as  those  on 
page  201,  distributed  throughout  the  book.  After  a  process  has  been 
studied  for  a  short  time,  tests  should  be  given  in  order  to  see  what 
progress  has  been  made,  to  check  the  effectiveness  of  the  teaching, 
and  to  reveal  difficulties  to  both  teacher  and  pupil.  It  is  suggested 
that  the  teacher  devise  and  give  brief  surprise  tests51  (not  over  ten 
minutes)  when,  the  following  pages  are  reached:  p.  161,  165,  174, 
179,  184,  189,  193,  197,  201,  206,  212,  213,  215,  218,  223,  224,  229, 
234,  239,  253,  265,  278,  281,  and  304. 

Use  of  standardized  tests.  Although  it  is  possible  for  all  testing 
to  be  done  by  means  of  tests  constructed  by  the  teacher,  standard- 
ized tests  offer  the  advantages  of  comparative  scores,  thoroughness  of 
diagnosis,  and  so  forth,  which  the  former  do  not  have.  However,  the 
justification  for  the  administration  of  standardized  tests  depends  upon 
the  uses  made  of  the  results.  If  such  tests  are  decided  upon,  they 
should  be  used  systematically  and  in  accordance  with  the  following 
suggestions: 

Early  in  the  year,  preferably  as  soon  as  the  class  is  well  started, 
the  pupils  are  familiar  with  the  teacher,  and  their  acquaintance  with 
arithmetic  is  renewed,  Monroe's  General  Survey  Scales  in  Arithmetic, 
Scale  I,  either  form,  should  be  given. 

For  purposes  of  diagnosis,  Monroe's  Diagnostic  Tests  in  Arith- 
metic, Parts  I  and  II,  and  Monroe's  Standardized  Reasoning  Tests 
in  Arithmetic  should  be  administered  near  the  end  of  the  first 
semester.   Remedial  work  based  upon  the  results  should  be  given.52 

As  an  aid  in  determining  promotion,  Monroe's  General  Survey 

Scales  in  Arithmetic,  Scale  I,  a  form  different  from  that  used  at  the 

beginning  of  the  year,  should  be  given  near  the  close.    The  pupils 

should  achieve  the  October  norm   for  the   fifth  grade.    The   other 

objectives  set  forth  on  pages  23-26  should  also  be  achieved. 

51For  a  good  discussion  of  the  general  principles  of  testing,  see: 
Monroe,  Walter   Scott,  DeVoss,  James  Clarence,   and  Kelly,  Frederick 
James.    Educational  Tests  and  Measurements.    Boston:    Houghton  Mifflin  Company, 
1924,  p.  469-86. 

"For  a  discussion  of  these  tests  and  the  use  to  make  of  the  results,  see: 
Ibid.,  p.  41-49,  58-64,  and  68-89. 

[36] 


STANDARDS  OF  ATTAINMENT  IN  THE  FUNDAMENTAL 
PROCESSES53 

The  following  standards  for  each  grade  are  taken  from  the  field 
of  standardized  tests,  namely,  the  Monroe  Diagnostic  Tests  in  Arith- 
metic and  the  Courtis  Standard  Research  Tests,  Series  B.  They  in- 
dicate what  the  average  child  should  be  able  to  accomplish  at  the  end 
of  each  grade.  For  example,  a  child  of  the  fourth  grade  is  supposed 
to  perform  correctly  a  certain  number  of  examples  similar  to  the 
ones  illustrated  for  his  grade  within  the  time  limits  specified.  These 
standards  or  objectives  are  definite.  There  is  no  doubt  in  the  mind 
of  the  teacher  as  to  what  her  pupils  of  average  intelligence  should  be 
able  to  accomplish.  At  present,  no  definite  standards  have  been 
determined  for  Grades  I,  II,  and  III,  but  those  for  the  remaining 
grades  of  the  elementary  school  are  given. 

Fourth-grade  standards.  A  fourth-grade  child  should  perform 
correctly  the  following  types  of  examples  at  the  following  rates : 

A.  Addition. 

Example  Number  Time 

4 

7  16  1  min. 

_2 

7 
6 
6 
5 
0 
5 


4  min. 


927 
379 
756 
837 
924 
110 
854 
965 
344 


B3Ruth  Streitz,  formerly  Associate,  Bureau  of  Educational  Research,   should 
be  given  the  credit  for  bringing  these  standards  together. 

[37] 


B.  Subtraction 

Example 

Number 

Time 

37 
_5_ 

1 

1  min. 

739 
367 

4 

1  min. 

107795491 
77197029 

7 

4  min. 

C.  Multiplication 

Example 

Number 

Time 

6572 
6 

2 

1  min. 

8246 
29 

1 

3  min. 

D.  Division 

Example 

Number 

Time 

8)3840 

1 

1  min. 

25)6775  2  4  min. 

Fifth-grade    standards.      A  fifth-grade   pupil   should   perform 
correctly  the  following  types  of  examples,  at  the  following  rates: 

A.  Addition 

Example  Number  Time 

2 

6  24  1  min. 

7 


4  min. 


297 
925 
473 
983 
315 
661 
794 
177 
124 


[38] 


8  min. 


B. 

Subtraction 

Example 
41 
8 

Number 
14 

Time 
1  min. 

508 
447 

6 

1  min. 

75088824 
57406394 

9 

4  min. 

C. 

Multiplication 

Example 
5862 

2 

Number 
4 

Time 
1  min. 

560 

37 

4 

4  min. 

6942 

58 

5 

6  min. 

D. 

Division 

Example 

Number 

Time 

4)7432 

2 
3 
6 

1  min. 

43)1591 

4  min. 

94)853252 

8  min. 

E. 

Fractions 

Example 
1/6  +  1/3 

Number 
4 

Time 
3  min. 

3/4  -  2/5 

3 

4  min. 

2/3  X3/4 

7 

2  min. 

2/5  +  1/3 

5 

4  min. 

Sixth-grade  standards.  A  sixth-grade  pupil  should  perform  cor- 
rectly the  following  types  of  examples  at  the  following  rates : 

A.  Addition 

Number 


Example 
8 
0 

5 


26 


Time 
1  min. 


[39] 


4  min. 


Example 

Number 

Time 

136 

340 

988 

386 

353 

10 

8  min. 

904 

547 

192 

439 

B. 

Subtraction 

Example 

Number 

Time 

53 

18 

1  min. 

9 

962 

8 

1  min. 

325 

91050005 

11 

4  min. 

19901563      ' 

C. 

Multiplication 

Example 

Number 

Time 

2845 

5 

1  min. 

8 

690 

8 

4  min. 

70 

5379 

8 

6  min. 

85 

D. 

Division 

Example 

Number 

Time 

6)4680 

6 
8 

2  min. 

37)9990 

8  min. 

E. 

Common  fractions 

Example 

Number 

Time 

3/10  +  2/5 

5 

3  min. 

5/6  -  3/4 

3 

4  min. 

2/5  X  3/7 

5 

1  min. 

4/7  +  2/3 

6 

4  min. 

F.  Decimal  fractions 

Multiplication:    The  decimal  point  is  to  be  placed  correctly  in  the  products 
already  given. 


657.2 
.7 

46400 

932.7 
.08 

74616 


Example 
82.74 
.4 

33096 

67.50 
.03 

20250" 


5.863 
.6 

35178 

8.409 
.07 

58863 


Number 


24 


Time 


1  min. 


[40] 


4065. 
5.1 


967.5 
8.4 


Example 


207315    712700 


7486. 
.76 


907.2 
.39 


14.53 
6.2 

90086 

61.32 

.17 

558936   353808 


104244 


8.637 
1.6 

138192 


2.893 
.68 

196724 


Number 


25 


Time 


1  min. 


Division:     The  answers  are  given  without  the  decimal  point.   Each  answer  is  to 
be  written  in  its  proper  position  and  the  decimal  point  inserted  in  its  proper  place. 


Example 
.4)148        Ans.: 

37 

.9)65.7 

Ans.: 

73 

.2)7.92 

Ans.: 

396 

.7). 301 

Ans.: 

43 

.03)16.2 

Ans.: 

54 

.06)7.44 

Ans.: 

124 

.02).  144 

Ans.: 

Ans.: 
Ans.: 
Ans.: 
Ans.: 
Ans.: 
Ans.: 

72 

.43)1591. 

37 

.63)35.91 

57 

2.1)140.7 

67 

2.8)21.980 

785 

83J531.2 

64 

79J36.893 

467 

Number 


Time 


1  min. 


1  min. 


2  min. 


Seventh-grade  standards.  A  seventh-grade  pupil  should  perform 
correctly  the  following  types  of  examples  at  the  following  rates : 

A.  Addition 

Number 


B.  Subtraction 


Example 
486 
765 
524 
140 
812 
466 
355 
834 
567 


Example 
87939983 
72207316 


11 


Time 


8  min. 


Number 
12 


Time 
4 


[41] 


C.  Multiplication 


Example 

2648 
46 


Number 
10 


Time 
6  min. 


D.  Division 


Example 


86)80066 


Number 
4 


Time 

3  min. 


E.  Common  fractions 


Example 

Number 

Time 

4/5  +  7/10 

1 

3  min. 

8/15  -  4/9 

5 

4  min. 

4/5  x7/9 

13 

2  min. 

4/5  4-  1-/2 

10 

4  min. 

F.  Decimal  fractions 

Multiplication:    The  decimal  point  is  to  be  placed  correctly  in  the  products 


already  given. 

Example 

657.2 
.7 

46400 

82.74 
.4 

33096 

5.863 
.6 

35178 

932.7 
.08 

74616 

67.50 
.03 

20250' 

8.409 

.07 

58863 

4065. 
5.1 

967.5 
8.4 

14.53 
6.2 

8.637 
1.6 

207315 

712700 

90086 

138192 

7486. 
.76 

907.2 
.39 

61.32 
.17 

2.893 
.68 

Number 


23 


Time 


1  min. 


22 


558936 


353808 


104244        196724 


Division:   The  answers  are  given  without  the  decimal  point.    Each  answer  is  to 
be  written  in  its  proper  position  and  the  decimal  point  inserted  in  its  proper  place. 


Example 
.4)148        Ans.: 
.9)65.7      Ans.: 
.2)7.92      Ans.: 
.7).  301       Ans.: 


Number 


37 

73 

396 

43 


Time 


1  min. 


[42] 


Example 

.03)16.2 

Ans.: 

54 

.06)7.44 

Ans. : 

124 

.02). 144 

Ans.: 

Ans.: 
Ans.: 

72 

.43)1591. 

37 

.63)35.91 

57 

2.1)140.7 

Ans.: 

67 

2.8)21.980 

Ans.: 
Ans.: 
Ans.: 

785 

83.)531.2 

64 

79. )36. 893 

467 

Number 


Time 


1  min. 


Eighth-grade  standards.   An  eighth-grade  pupil  should  perform 
correctly  the  following  types  of  examples  at  the  following  rates: 

A.  Addition 

Number 


Example 


Time 


783 
697 
200 
366 
851 
535 
323 
229 

12 

8  min. 

B. 

Subtraction 

Example 

Number 

Time 

160670971 
80361837 

13 

4  min. 

C. 

Multiplication 

Example 

Number 

Time 

4263 

37 

11 

6  min. 

D. 

Division 

Example 

Number 

5 

Time 

73)58765 

8  mir 

E. 

Common  fractions 

Example 

Number 

Time 

5/8  +  3/4 

10 

3  min. 

4/5  -  1/3 

7 

4  min. 

1/6  x  3/10 

15 

2  min. 

5/12  --  4/9 

7  . 

2  min. 

[43] 


F.  Decimal  fractions 


Multiplication:    The  decimal  point  is  to  be  placed  correctly  in  the  products 
already  given. 


558936 


Example 


657.2 

.7 

82.74 
.4 

5.863 
.6 

46400 

33096 

35178 

932.7 
.08 

74616 

67.50 
.03 

20250 

8.409 
.07 

58863 

4065. 
5.1 

967.5 
8.4 

14.53 
6.2 

8.637 
1.6 

207315 

712700 

90086 

138192 

7486. 
.76 

907.2 
.39 

61.32 
.17 

2.893 
.68 

353808 


104244 


196724 


Number 


26 


27 


Time 


Division:   The  answers  are  given  without  the  decimal  point.    Each  answer  is  to 
be  written  in  its  proper  position  and  the  decimal  point  inserted  in  its  proper  place. 


Example 

.4)148 

Ans.: 

37 

.9)65.7 

Ans.: 

73 

.2)7.92 

Ans.: 

396 

.7). 301 

Ans.: 

43 

.03)16.2 

Ans.: 

54 

.06)7.44 

Ans.: 

124 

.02). 144 

Ans.: 

Ans.: 
Ans. : 
Ans.: 
Ans.: 
Ans.: 

72 

.43)1591. 

37 

.63)35.91 

57 

2.1)140.7 

67 

2.8)21.980 

785 

83.)531.2 

64 

Number 


Time 


1  min. 


1  min. 


79.)36.893       Ans.:       467 


[44] 


SELECTED  AND  ANNOTATED  BIBLIOGRAPHY 
Introductory  statement.  This  bibliography  is  made  up  of  refer- 
ences that  will  be  helpful  to  those  who  make  courses  of  study  in 
arithmetic.  No  attempt  has  been  made  to  include  all  possible  refer- 
ences. The  bibliography  is  divided  into  five  groups:  first,  general 
references  on  curriculum  and  course-of-study  making;  second,  books 
and  articles  on  methods  of  teaching  arithmetic;  third,  arithmetic 
courses  of  study;  fourth,  references  on  testing  and  standards  of 
achievement;  and  fifth,  reports  of  investigations  and  miscellaneous 
references. 

I.    GENERAL  REFERENCES  ON  CURRICULUM  AND 
COURSE-OF-STUDY  MAKING 

Bobbitt,  Franklin.   How  to  Make  a  Curriculum.    Boston:    Hough- 
ton Mifflin  Company,   1924,  p.   1-75. 

This  is  a  report  of  the  work  on  revising  the  curriculum  in  Los  Angeles,  which 
Dr.  Bobbitt  directed  over  a  period  of  two  years. 

Caldwell,  Otis  W.    "Types  and  principles  of  curricular  develop- 
ment."  Teachers  College  Record,  24:326-37,  September,  1923. 

Speech  delivered  at  meeting  of  the  Department  of  Superintendence  of  the 
National  Education  Association  at  Cleveland,  February  28,  1923.  Outlines  the 
method  and  results  of  two  types  of  curricular  investigations  and  states  certain 
principles  for  use  in  reorganizing  school  subjects  of  study. 

Charters,  W.  W.   Curriculum  Construction.   New  York:    The  Mac- 
millan  Company,  1923,  p.  3-168. 

This  portion  of  the  book  gives  a  good  background  theory  of  curriculum  con- 
struction and  presents  Dr.  Charters'  own  point  of  view. 

McMurry,   Charles  A.    How  to  Organize  the   Curriculum.    New 
York:    The  Macmillan  Company,  1923.    358  p. 

The  curriculum  is  discussed  in  terms  of  projects,  type  studies,  and  large 
units  of  study.  A  suggested  curriculum  of  large  teaching  units  is  given,  covering  the 
fields  of  geography,  history,   science,   and  literature. 

Monroe,  Walter  S.    "Making  a  course  of  study."    University  of 

Illinois  Bulletin,  Vol.  23,  No.  2,  Bureau  of  Educational  Research 

Circular  No.  35.  Urbana:    University  of  Illinois,  1925.   35  p. 

This  circular  presents  the  best  present  day  ideas  on  the  general  make-up  of 
courses  of  study,  the  way  to  go  about  making  a  course  of  study,  and  the  benefits  to 
be  derived^  from  such  work.  A  lengthy  bibliography  on  curriculum  and  course-of- 
study  making  is  included. 

[45] 


Threlkeld,  A.  L.  "Curriculum  revision:  how  a  particular  city  may 
attack  the  problem,"  Elementary  School  Journal,  25:573-82, 
April,  1925. 

This  is  a  report  of  the  method  of  attack  used  in  Denver,  Colorado. 

Wilson,  H.  B.    "The  course  of  study  in  the  work  of  the  modern 

school,"  Course  of  Study  Monographs,  Introductory.    Berkeley, 

California:    Board  of  Education,  1921.    14  p. 

"Introductory  to  all  (Berkeley)  Courses  of  Study  presenting  the  general  point 
of  view  which  has  guided  the  formulation  of  the  detailed  course  in  all  subjects  for 
the  various  schools."   (Introductory  Note.) 

"The  elementary  school  curriculum."  Second  Yearbook  of  the  De- 
partment of  Superintendence,  Washington:  Department  of 
Superintendence  of  the  National  Education  Association,  1924. 
296  p. 

A  fair  presentation  of  the  elementary  curriculum  situation  in  the  United  States 
in  1923  is  given. 

"Facts  on  the  public  school  curriculum."  Research  Bulletin  of  the 
National  Education  Association,  Vol.  I,  No.  5.  Washington: 
Research  Division  of  the  National  Education  Association,  1923, 
p.  310-50. 

This   bulletin    furnishes    good    source    material    on    time    allotments,    statutory 
requirements,  grade  combinations  of  subjects,  and  other  pertinent  matters. 

II.  METHODS  OF  TEACHING  ARITHMETIC 

Brown,  Joseph  C.  and  Coffman,  Lotus  D.  How  to  Teach  Arith- 
metic.   Chicago:    Row,  Peterson  and  Company,  1914.   373  p. 

Although  more  than  ten  years  old,  this  book  is  not  at  all  out  of  date  either 
in  point  of  view  or  in  helpful  suggestions. 

Charters,    W.    W.     Teaching    the    Common    Branches.     Boston: 

Houghton  Mifflin  Company,  1913,  p.  278-99. 

The   pages   selected   contain   a  particularly  valuable   discussion   on   the  use  of 
textbooks. 

Klapper,  P.   The  Teaching  of  Arithmetic.   New  York:    D.  Appleton 

and  Company,  1921.   393  p. 

One  of  the  best  books  on  methods  of  teaching  arithmetic. 
Lennes,  N.  J.   The  Teaching  of  Arithmetic.   New  York:    The  Mac- 

millan  Company,  1923.   486  p. 

A  stimulating  book  combining  theory  with  illustrations. 


[46] 


Monroe,  Walter  S.  "Principles  of  method  in  teaching  arithmetic,  as 
derived  from  scientific  investigations,"  Eighteenth  Yearbook  of 
the  National  Society  for  the  Study  of  Education,  Part  II. 
Bloomington,  Illinois:  Public  School  Publishing  Company,  1919, 
p.  78-95. 

Twenty-five    fundamental    principles    of    method    are    stated    and    briefly    dis- 
cussed. 

Osburn,  Worth  J.  Corrective  Arithmetic.  Boston:  Houghton 
Mifflin  Company,  1924.    182  p. 

A  rather  comprehensive  treatment  of  pupils'  difficulties  and  ways  of  meeting 
them. 

Streitz,  Ruth.  "Teachers'  difficulties  in  arithmetic  and  their  cor- 
rectives." University  of  Illinois  Bulletin,  Vol.  21,  No.  34,  Bureau 
of  Educational  Research  Bulletin  No.  18.  Urbana:  University 
of  Illinois,  1924.   34  p. 

Twenty-eight  difficulties  are  listed  with  correctives  that   are  actually  in   suc- 
cessful use. 

Thorndike,  Edward  Lee.  The  New  Methods  in  Arithmetic. 
Chicago:   Rand  McNally  and  Company,  1921.  260  p. 

Abundant   detailed  illustrations  and   applications  are   given   for  the  principles 
discussed. 


III.    COURSES  OF  STUDY  IN  ARITHMETIC 

"Arithmetic."    Course   of   Study   Monograph,   Elementary   Schools, 

No.  1.   Berkeley,  California:    Board  of  Education,  1921.  91  p. 

Course  of   study   for  the  first   six  grades,   divided  into   semesters.     Especially 
good  on  methods  and  learning  exercises. 

"Arithmetic— course  of  study  for  grades  four,  five,  and  six."  Balti- 
more, Maryland:    Department  of  Education,  1924.    Ill  p. 

Especially  helpful  in  outlining  of  subject-matter  and  in  suggestions  on  methods 
of  teaching. 

"Arithmetic — elementary  course  of  study."    Trenton,  New  Jersey: 
Board  of  Education,  1923.   96  p. 

Some  good  suggestions  on  provisions  for  individual  differences  and  on  general 
objectives. 

"Arithmetic— grades    1,  2,  3,  4,   5,   and  6— course  of  study   mono- 
graph."   Denver:    Board  of  Education,  1924.    228  p. 

Probably  the  most  comprehensive  course  of  study  in  arithmetic  yet  published. 
Very  suggestive. 


[47] 


"Arithmetic — syllabus   for   elementary   schools."    University   of  the 

State  of  New  York  Bulletin,  No.  815.    Albany:    University  of 

the  State  of  New  York  Press,  1925.    121  p. 

This  is  not  a  course  of  study  but  is  only  a  syllabus,  containing  both  con- 
densed and  expanded  outlines  for  the  eight  elementary  grades.  Many  helpful  sug- 
gestions on  methods  of  teaching  are  given  throughout  the  expanded  outlines. 

"Course  of  study,  public  schools,  Baltimore  County,  Maryland, 
Grades  I-VIII."  Baltimore,  Maryland:  Warwick  and  York, 
1921,  p.  261-329. 

The  parts  for  the  first  four  grades  are  the  best.  Most  helpful  in  the  matter 
of  time  allotments  within  recitations  and  games. 

"Geography — history — arithmetic — course  of  study  for  kindergarten 
and  grades  one,  two  and  three."  Baltimore,  Maryland:  Depart- 
ment of  Education,  1924.  78  p. 

This  contains  the  portion  of  the  course  of  study  in  arithmetic  which  precedes 
that  given  previously  in  this  bibliography. 

"Long  Beach  City  Schools  course  of  study."  Long  Beach,  California: 

Board  of  Education,  1924.    (Five  separate  monographs  for  the 

first  six  grades,  having  from  19  to  40  pages  each.) 

These  course-of-study  monographs  are  especially  good  in  making  provisions 
for  individual  differences  and  in  showing  what  the  course  of  study  can  do  by  way  of 
directing  teachers  in  the  use  of  the  textbook. 

IV.    TESTING  AND  STANDARDS  OF  ACHIEVEMENT 

Doherty,  Margaret  and  MacLatchy,  Josephine.    "Bibliography 

of    educational    and    psychological    tests    and    measurements." 

United    States    Bureau   of   Education    Bulletin,    1923,    No.    55. 

Washington:    Government  Printing  Office,  1924.    233  p. 

This  bibliography  gives  not  only  the  tests  but  a  rather  complete  list  of 
references  that  discuss  the  particular  tests,  the  uses  of  tests  in  general,  and  the  uses 
of  tests  according  to  types  of  schools. 

Monroe,  Walter  Scott,  DeVoss,  James  Clarence,  and  Kelly, 
Frederick  James.  Educational  Tests  and  Measurements.  (Re- 
vised Edition.)  Boston:  Houghton  Mifflin  Company,  1924, 
p.  1-93,  417-30,  469-86. 

The  structure,  uses,  and  limitations  of  most  of  the  standardized  tests  in  arith- 
metic are  discussed  in  the  sections  referred  to.  The  general  theory  of  testing  is 
discussed  and  practical  suggestions  made.  An  excellent  bibliography  on  testing  in 
arithmetic  is  given  on  pages  91-93. 


[48] 


Odell,  Charles  W.  "Educational  tests  for  use  in  elementary 
schools,  revised."  University  of  Illinois  Bulletin,  Vol.  22,  No.  16. 
Bureau  of  Educational  Research  Circular  No.  33.  Urbana:  Uni- 
versity of  Illinois,  1924.   22  p. 

An  annotated  bibliography  of  tests  that  are  now  available.    "Tests  that  are 

known  to  be  distinctly  unsatisfactory  are  omitted  .  .  ."    Norms  are   available  for 

most  of  the  tests  listed.    The  bibliography  is  preceded  by  a  brief  discussion  of  the 
characteristics  and  use  of  tests. 

Bureau  of  Cooperative  Research  (Compiled  by).  First  Revision  of 
Bibliography  of  Educational  Measurements.  Bulletin  of  the 
School  of  Education,  Vol.  1,  No.  5.  Bloomington,  Indiana: 
Indiana  University,  1925.     147  p. 

"This  bibliography  is  compiled  for  the  double  purpose  of  listing  all  efforts,  so  far 
as  they  have  come  to  our  attention,  that  have  been  made  in  the  United  States  to 
develop  achievement  tests,  and  of  giving  a  brief  description  of  each  test,  including 
in  the  description  not  only  an  analysis  of  the  test  and  its  purpose  but  also  available 
information  concerning  the  range  of  the  test,  administration  cost  of  the  test  in 
the  publisher  of  the  test,  and  the  date  of  publication."  (From  foreword  of  first 
edition.) 

V.  REPORTS  OF  INVESTIGATIONS  AND  MISCELLANEOUS 

REFERENCES 

Buswell,  Guy  Thomas  and  Judd,  Charles  Hubbard.  "Summary 
of  educational  investigations  relating  to  arithmetic."  Supplemen- 
tary Educational  Monographs,  No.  27.  Chicago:  University  of 
Chicago,  1925.   212  p. 

A  most  valuable  and  usable  summary  of  307  articles  and  books  of  merit  which 
report  scientific  investigations  of  the  methods  and  results  of  teaching  arithmetic. 

Davis,  M.  Elsie.    "The  development  of  the  fundamental  number 

habits,"  The  School  Magazine,  4:48-50,  October,  1921. 

Published  by  the  Board  of  Education  of  Buffalo,  New  York.  Contains  sug- 
gestions on  association  and  learning  of  the  one  hundred  addition  and  one  hundred 
subtraction  facts. 

Davis,  Roy.  "Business  practice  in  elementary  schools."  Harvard 
Bulletins  in  Education,  Vol.  6,  Cambridge:  Harvard  Univer- 
sity, October,  1917. 

Report  on  an  investigation  as  to  what  knowledge  of  business  terms  (money, 
credit,  interest,  and  so  forth)  is  had  by  children  in  the  elementary  school.  Con- 
tains suggestions  on  teaching  arithmetic  with  these  terms  in  view. 

Jessup,  Walter  A.  "Current  practices  and  standards  in  arithmetic," 
Fourteenth  Yearbook  of  the  National  Society  for  the  Study  of 
Education,  Part  I.  Bloomington,  Illinois:  Public  School  Pub- 
lishing Company,  1915,  p.  116-30. 

[49] 


Makes  recommendations,  based  on  judgments  and  practices  of  superintendents, 
as  to  elimination  of  topics,  increased  emphasis,  time  devoted  to  recitations,  grade 
occurrence  of  topics,  and  objective  standards. 

Monroe,  Walter  S.  "A  preliminary  report  of  an  investigation  of  the 
economy  of  time  in  arithmetic,"  Sixteenth  Yearbook  of  the 
National  Society  for  the  Study  of  Education,  Part  I.  Blooming- 
ton,  Illinois:  Public  School  Publishing  Company,  1917,  p. 
111-27. 

Especially  valuable  because  of  the  explicit  recognition  of  different  types  of 
problems. 

Wilson,  Guy  M.  "A  survey  of  the  social  and  business  uses  of  arith- 
metic," Sixteenth  Yearbook  of  the  National  Society  for  the 
Study  of  Education,  Part  I.  Bloomington,  Illinois:  Public 
School  Publishing  Company,  1917,  p.  128-42. 

This  is  a  report  of  one  of  the  foremost  investigations  that  points  out  the 
influence  on  the  arithmetic  curriculum  of  business  and  social  usage. 

Wilson,  Guy  M.  "Arithmetic,"  Third  Yearbook  of  the  Department 
of  Superintendence.  Washington:  Department  of  Superintend- 
ence of  the  National  Education  Association,  1925,  p.  35-109. 

A  digest  of  most  of  the  special  studies  which  have  been  made  on  the  arith- 
metic curriculum. 

"Material   for   arithmetical   problems."     Department   of   Education, 

Division  of  Reference  and  Research,  Bulletin  No.  2.   New  York 

City:    Board  of  Education,  1914. 

This  bulletin  consists  of  many  arithmetical  problems,  which  are  the  result  of 
a  study  made  in  New  York  City  at  the  suggestion  of  S.  A.  Courtis,  one  of  the 
members  of  the  School  Inquiry  Committee.  There  is  also  a  good  discussion  of  the 
nature  of  the  arithmetical  problem. 

On  minimum  essentials:  Fourteenth  Yearbook  of  the  National 
Society  for  the  Study  of  Education,  Part  I;  Sixteenth  Yearbook, 
Part  I;  Seventeenth  Yearbook,  Part  I;  and  Eighteenth  Year- 
book, Part  II.  Bloomington,  Illinois:  Public  School  Publishing 
Company,  1915,  1917,  1918,  1919. 

These  are  reports  on  economy  of  time  and  minimal  essentials  in  elementary 
school  subjects.  The  discussions  furnish  a  background  for  such  provisions  in 
courses  of  study.    Some  concrete  material  and  suggestions  are  also  provided. 


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